Multiset - Wikipedia

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<script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script> <div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"> <section class="mf-section-0" id="mf-section-0"> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style> <div role="note" class="hatnote navigation-not-searchable"> This article is about the mathematical concept. For the computer science data structure, see <a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset_(abstract_data_type)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Multiset (abstract data type)">Multiset (abstract data type)</a>. </div> In <a href="https://en-m-wikipedia-org.translate.goog/wiki/Mathematics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mathematics">mathematics</a>, a multiset (or bag, or mset) is a modification of the concept of a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Set_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Set (mathematics)">set</a> that, unlike a set,<a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Cantor-1">[1]</a> allows for multiple instances for each of its <a href="https://en-m-wikipedia-org.translate.goog/wiki/Element_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Element (mathematics)">elements</a>. The number of instances given for each element is called the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiplicity_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Multiplicity (mathematics)">multiplicity</a> of that element in the multiset. As a consequence, an infinite number of multisets exist that contain only elements a and b, but vary in the multiplicities of their elements: <ul> <li>The set {a, b} contains only elements a and b, each having multiplicity 1 when {a, b} is seen as a multiset.</li> <li>In the multiset {a, a, b}, the element a has multiplicity 2, and b has multiplicity 1.</li> <li>In the multiset {a, a, a, b, b, b}, a and b both have multiplicity 3.</li> </ul> These objects are all different when viewed as multisets, although they are the same set, since they all consist of the same elements. As with sets, and in contrast to <a href="https://en-m-wikipedia-org.translate.goog/wiki/Tuple?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Tuple">tuples</a>, the order in which elements are listed does not matter in discriminating multisets, so {a, a, b} and {a, b, a} denote the same multiset. To distinguish between sets and multisets, a notation that incorporates square brackets is sometimes used: the multiset {a, a, b} can be denoted by [a, a, b].<a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-2">[2]</a> The <a href="https://en-m-wikipedia-org.translate.goog/wiki/Cardinality?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Cardinality">cardinality</a> or "size" of a multiset is the sum of the multiplicities of all its elements. For example, in the multiset {a, a, b, b, b, c} the multiplicities of the members a, b, and c are respectively 2, 3, and 1, and therefore the cardinality of this multiset is 6. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Nicolaas_Govert_de_Bruijn?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Nicolaas Govert de Bruijn">Nicolaas Govert de Bruijn</a> coined the word multiset in the 1970s, according to <a href="https://en-m-wikipedia-org.translate.goog/wiki/Donald_Knuth?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Donald Knuth">Donald Knuth</a>.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Knuth1998-3">[3]</a>: 694 However, the concept of multisets predates the coinage of the word multiset by many centuries. Knuth himself attributes the first study of multisets to the Indian mathematician <a href="https://en-m-wikipedia-org.translate.goog/wiki/Bh%C4%81skara_II?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Bhāskara II">Bhāskarāchārya</a>, who described <a href="https://en-m-wikipedia-org.translate.goog/wiki/Permutation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Permutations_of_multisets" title="Permutation">permutations of multisets</a> around 1150. Other names have been proposed or used for this concept, including list, bunch, bag, heap, sample, weighted set, collection, and suite.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Knuth1998-3">[3]</a>: 694 <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"> <input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"> <div class="toctitle" lang="en" dir="ltr"> <h2 id="mw-toc-heading">Contents</h2><label class="toctogglelabel" for="toctogglecheckbox"></label> </div> <ul> <li class="toclevel-1 tocsection-1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#History">1 History</a></li> <li class="toclevel-1 tocsection-2"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Examples">2 Examples</a></li> <li class="toclevel-1 tocsection-3"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Definition">3 Definition</a></li> <li class="toclevel-1 tocsection-4"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Basic_properties_and_operations">4 Basic properties and operations</a></li> <li class="toclevel-1 tocsection-5"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Counting_multisets">5 Counting multisets</a> <ul> <li class="toclevel-2 tocsection-6"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Recurrence_relation">5.1 Recurrence relation</a></li> <li class="toclevel-2 tocsection-7"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Generating_series">5.2 Generating series</a></li> <li class="toclevel-2 tocsection-8"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Generalization_and_connection_to_the_negative_binomial_series">5.3 Generalization and connection to the negative binomial series</a></li> </ul></li> <li class="toclevel-1 tocsection-9"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Applications">6 Applications</a></li> <li class="toclevel-1 tocsection-10"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Generalizations">7 Generalizations</a></li> <li class="toclevel-1 tocsection-11"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#See_also">8 See also</a></li> <li class="toclevel-1 tocsection-12"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Notes">9 Notes</a></li> <li class="toclevel-1 tocsection-13"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#References">10 References</a></li> </ul> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"> <h2 id="History">History</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Multiset&action=edit&section=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: History" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-1 collapsible-block" id="mf-section-1"> Wayne Blizard traced multisets back to the very origin of numbers, arguing that "in ancient times, the number n was often represented by a collection of n strokes, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Tally_mark?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Tally mark">tally marks</a>, or units."<a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Blizard1989-4">[4]</a> These and similar collections of objects can be regarded as multisets, because strokes, tally marks, or units are considered indistinguishable. This shows that people implicitly used multisets even before mathematics emerged. Practical needs for this structure have caused multisets to be rediscovered several times, appearing in literature under different names.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Blizard1991-5">[5]</a>: 323 For instance, they were important in early <a href="https://en-m-wikipedia-org.translate.goog/wiki/AI?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="AI">AI</a> languages, such as QA4, where they were referred to as bags, a term attributed to <a href="https://en-m-wikipedia-org.translate.goog/wiki/L._Peter_Deutsch?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="L. Peter Deutsch">Peter Deutsch</a>.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Rulifson1972-6">[6]</a> A multiset has been also called an aggregate, heap, bunch, sample, weighted set, occurrence set, and fireset (finitely repeated element set).<a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Blizard1991-5">[5]</a>: 320 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Singh2007-7">[7]</a> Although multisets were used implicitly from ancient times, their explicit exploration happened much later. The first known study of multisets is attributed to the Indian mathematician <a href="https://en-m-wikipedia-org.translate.goog/wiki/Bh%C4%81skara_II?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Bhāskara II">Bhāskarāchārya</a> circa 1150, who described permutations of multisets.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Knuth1998-3">[3]</a>: 694 The work of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Marius_Nizolius?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Marius Nizolius">Marius Nizolius</a> (1498–1576) contains another early reference to the concept of multisets.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Angelelli1965-8">[8]</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Athanasius_Kircher?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Athanasius Kircher">Athanasius Kircher</a> found the number of multiset permutations when one element can be repeated.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-9">[9]</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Jean_Prestet?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Jean Prestet">Jean Prestet</a> published a general rule for multiset permutations in 1675.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-10">[10]</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/John_Wallis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="John Wallis">John Wallis</a> explained this rule in more detail in 1685.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-11">[11]</a> Multisets appeared explicitly in the work of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Richard_Dedekind?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Richard Dedekind">Richard Dedekind</a>.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-12">[12]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-syropoulos-13">[13]</a> Other mathematicians formalized multisets and began to study them as precise mathematical structures in the 20th century. For example, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Hassler_Whitney?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Hassler Whitney">Hassler Whitney</a> (1933) described generalized sets ("sets" whose <a href="https://en-m-wikipedia-org.translate.goog/wiki/Characteristic_function?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Characteristic function">characteristic functions</a> may take any <a href="https://en-m-wikipedia-org.translate.goog/wiki/Integer?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Integer">integer</a> value: positive, negative or zero).<a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Blizard1991-5">[5]</a>: 326 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Whitney-14">[14]</a>: 405 Monro (1987) investigated the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Category_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Category (mathematics)">category</a> Mul of multisets and their <a href="https://en-m-wikipedia-org.translate.goog/wiki/Morphism?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Morphism">morphisms</a>, defining a multiset as a set with an <a href="https://en-m-wikipedia-org.translate.goog/wiki/Equivalence_relation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Equivalence relation">equivalence relation</a> between elements "of the same sort", and a morphism between multisets as a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Function_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Function (mathematics)">function</a> that respects sorts. He also introduced a multinumber : a function f (x) from a multiset to the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Natural_number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Natural number">natural numbers</a>, giving the multiplicity of element x in the multiset. Monro argued that the concepts of multiset and multinumber are often mixed indiscriminately, though both are useful.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Blizard1991-5">[5]</a>: 327–328 <a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-15">[15]</a> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"> <h2 id="Examples">Examples</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Multiset&action=edit&section=2&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Examples" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-2 collapsible-block" id="mf-section-2"> One of the simplest and most natural examples is the multiset of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Prime_factor?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Prime factor">prime factors</a> of a natural number n. Here the underlying set of elements is the set of prime factors of n. For example, the number <a href="https://en-m-wikipedia-org.translate.goog/wiki/120_(number)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="120 (number)">120</a> has the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Prime_factorization?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Prime factorization">prime factorization</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 120=2^{3}3^{1}5^{1},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn> 120 </mn> <mo> = </mo> <msup> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> <msup> <mn> 3 </mn> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msup> <msup> <mn> 5 </mn> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msup> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle 120=2^{3}3^{1}5^{1},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c629f4c3d9baa2eabbace9e09ce3c4a7e0286539" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.883ex; height:3.009ex;" alt="{\displaystyle 120=2^{3}3^{1}5^{1},}"> </noscript><span class="lazy-image-placeholder" style="width: 13.883ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c629f4c3d9baa2eabbace9e09ce3c4a7e0286539" data-alt="{\displaystyle 120=2^{3}3^{1}5^{1},}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  which gives the multiset {2, 2, 2, 3, 5}. A related example is the multiset of solutions of an <a href="https://en-m-wikipedia-org.translate.goog/wiki/Algebraic_equation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Algebraic equation">algebraic equation</a>. A <a href="https://en-m-wikipedia-org.translate.goog/wiki/Quadratic_equation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Quadratic equation">quadratic equation</a>, for example, has two solutions. However, in some cases they are both the same number. Thus the multiset of solutions of the equation could be {3, 5}, or it could be {4, 4}. In the latter case it has a solution of multiplicity 2. More generally, the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Fundamental_theorem_of_algebra?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Fundamental theorem of algebra">fundamental theorem of algebra</a> asserts that the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Complex_number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Complex number">complex</a> solutions of a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Polynomial_equation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Polynomial equation">polynomial equation</a> of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Degree_of_a_polynomial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Degree of a polynomial">degree</a> d always form a multiset of cardinality d. A special case of the above are the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Eigenvalue?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Eigenvalue">eigenvalues</a> of a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Matrix_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Matrix (mathematics)">matrix</a>, whose multiplicity is usually defined as their multiplicity as <a href="https://en-m-wikipedia-org.translate.goog/wiki/Root_of_a_polynomial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Root of a polynomial">roots</a> of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Characteristic_polynomial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Characteristic polynomial">characteristic polynomial</a>. However two other multiplicities are naturally defined for eigenvalues, their multiplicities as roots of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Minimal_polynomial_(linear_algebra)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Minimal polynomial (linear algebra)">minimal polynomial</a>, and the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Geometric_multiplicity?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Geometric multiplicity">geometric multiplicity</a>, which is defined as the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Dimension_(vector_space)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Dimension (vector space)">dimension</a> of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Kernel_(linear_algebra)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Kernel (linear algebra)">kernel</a> of A − λI (where λ is an eigenvalue of the matrix A). These three multiplicities define three multisets of eigenvalues, which may be all different: Let A be a n × n matrix in <a href="https://en-m-wikipedia-org.translate.goog/wiki/Jordan_normal_form?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Jordan normal form">Jordan normal form</a> that has a single eigenvalue. Its multiplicity is n, its multiplicity as a root of the minimal polynomial is the size of the largest Jordan block, and its geometric multiplicity is the number of Jordan blocks. </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"> <h2 id="Definition">Definition</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Multiset&action=edit&section=3&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Definition" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-3 collapsible-block" id="mf-section-3"> A multiset may be formally defined as an <a href="https://en-m-wikipedia-org.translate.goog/wiki/Ordered_pair?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ordered pair">ordered pair</a> (A, m) where A is the underlying set of the multiset, formed from its distinct elements, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\colon A\to \mathbb {Z} ^{+}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> m </mi> <mo> : </mo> <mi> A </mi> <mo stretchy="false"> → </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> Z </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> + </mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m\colon A\to \mathbb {Z} ^{+}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f0777fda3ca3afbfa3aa58c7df5221a924c0c8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.493ex; height:2.509ex;" alt="{\displaystyle m\colon A\to \mathbb {Z} ^{+}}"> </noscript><span class="lazy-image-placeholder" style="width: 11.493ex;height: 2.509ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f0777fda3ca3afbfa3aa58c7df5221a924c0c8d" data-alt="{\displaystyle m\colon A\to \mathbb {Z} ^{+}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is a function from A to the set of positive integers, giving the multiplicity – that is, the number of occurrences – of the element a in the multiset as the number m(a). (It is also possible to allow multiplicity 0 or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal"> ∞ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \infty } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }"> </noscript><span class="lazy-image-placeholder" style="width: 2.324ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26c105004f30c27aa7c2a9c601550a4183b1f21" data-alt="{\displaystyle \infty }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , especially when considering submultisets.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-16">[16]</a> This article is restricted to finite, positive multiplicities.) Representing the function m by its <a href="https://en-m-wikipedia-org.translate.goog/wiki/Graph_of_a_function?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Graph of a function">graph</a> (the set of ordered pairs <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(a,m(a)):a\in A\}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> { </mo> <mo stretchy="false"> ( </mo> <mi> a </mi> <mo> , </mo> <mi> m </mi> <mo stretchy="false"> ( </mo> <mi> a </mi> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mo> : </mo> <mi> a </mi> <mo> ∈ </mo> <mi> A </mi> <mo fence="false" stretchy="false"> } </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \{(a,m(a)):a\in A\}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6eec64e9171b64c01fa8d789f7c87204295005d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.228ex; height:2.843ex;" alt="{\displaystyle \{(a,m(a)):a\in A\}}"> </noscript><span class="lazy-image-placeholder" style="width: 19.228ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6eec64e9171b64c01fa8d789f7c87204295005d" data-alt="{\displaystyle \{(a,m(a)):a\in A\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ) allows for writing the multiset {a, a, b} as {(a, 2), (b, 1)}, and the multiset {a, b} as {(a, 1), (b, 1)}. This notation is however not commonly used; more compact notations are employed. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\{a_{1},\ldots ,a_{n}\}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> A </mi> <mo> = </mo> <mo fence="false" stretchy="false"> { </mo> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo fence="false" stretchy="false"> } </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle A=\{a_{1},\ldots ,a_{n}\}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04ad1d6d9c8b7851d73837fe9c160851abac4777" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.077ex; height:2.843ex;" alt="{\displaystyle A=\{a_{1},\ldots ,a_{n}\}}"> </noscript><span class="lazy-image-placeholder" style="width: 17.077ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04ad1d6d9c8b7851d73837fe9c160851abac4777" data-alt="{\displaystyle A=\{a_{1},\ldots ,a_{n}\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Finite_set?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Finite set">finite set</a>, the multiset (A, m) is often represented as <dl> <dd> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{a_{1}^{m(a_{1})},\ldots ,a_{n}^{m(a_{n})}\right\},\quad }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo> { </mo> <mrow> <msubsup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> <mo stretchy="false"> ( </mo> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> </mrow> </msubsup> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msubsup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> <mo stretchy="false"> ( </mo> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> </mrow> </msubsup> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mspace width="1em"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left\{a_{1}^{m(a_{1})},\ldots ,a_{n}^{m(a_{n})}\right\},\quad } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d537f9e55708d22f056ba6e445c11a35807a3b7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.54ex; height:4.843ex;" alt="{\displaystyle \left\{a_{1}^{m(a_{1})},\ldots ,a_{n}^{m(a_{n})}\right\},\quad }"> </noscript><span class="lazy-image-placeholder" style="width: 23.54ex;height: 4.843ex;vertical-align: -1.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d537f9e55708d22f056ba6e445c11a35807a3b7e" data-alt="{\displaystyle \left\{a_{1}^{m(a_{1})},\ldots ,a_{n}^{m(a_{n})}\right\},\quad }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  sometimes simplified to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad a_{1}^{m(a_{1})}\cdots a_{n}^{m(a_{n})},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em"></mspace> <msubsup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> <mo stretchy="false"> ( </mo> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> </mrow> </msubsup> <mo> ⋯ </mo> <msubsup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> m </mi> <mo stretchy="false"> ( </mo> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> </mrow> </msubsup> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \quad a_{1}^{m(a_{1})}\cdots a_{n}^{m(a_{n})},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1a18debd0ededc6515c7b4628ad99e3f6fc27c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.371ex; height:3.676ex;" alt="{\displaystyle \quad a_{1}^{m(a_{1})}\cdots a_{n}^{m(a_{n})},}"> </noscript><span class="lazy-image-placeholder" style="width: 18.371ex;height: 3.676ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1a18debd0ededc6515c7b4628ad99e3f6fc27c7" data-alt="{\displaystyle \quad a_{1}^{m(a_{1})}\cdots a_{n}^{m(a_{n})},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </dd> </dl> where upper indices equal to 1 are omitted. For example, the multiset {a, a, b} may be written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{a^{2},b\}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> { </mo> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> , </mo> <mi> b </mi> <mo fence="false" stretchy="false"> } </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \{a^{2},b\}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8779c4198c27967767f68825e7b721cdd7d737da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.64ex; height:3.176ex;" alt="{\displaystyle \{a^{2},b\}}"> </noscript><span class="lazy-image-placeholder" style="width: 6.64ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8779c4198c27967767f68825e7b721cdd7d737da" data-alt="{\displaystyle \{a^{2},b\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2}b.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mi> b </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle a^{2}b.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/748588f7899b95e4d696564b0dc1234f481158fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.928ex; height:2.676ex;" alt="{\displaystyle a^{2}b.}"> </noscript><span class="lazy-image-placeholder" style="width: 3.928ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/748588f7899b95e4d696564b0dc1234f481158fe" data-alt="{\displaystyle a^{2}b.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  If the elements of the multiset are numbers, a confusion is possible with ordinary <a href="https://en-m-wikipedia-org.translate.goog/wiki/Arithmetic_operations?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Arithmetic operations">arithmetic operations</a>; those normally can be excluded from the context. On the other hand, the latter notation is coherent with the fact that the prime factorization of a positive integer is a uniquely defined multiset, as asserted by the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Fundamental_theorem_of_arithmetic?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Fundamental theorem of arithmetic">fundamental theorem of arithmetic</a>. Also, a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Monomial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Monomial">monomial</a> is a multiset of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Indeterminate_(variable)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Indeterminate (variable)">indeterminates</a>; for example, the monomial x3y2 corresponds to the multiset {x, x, x, y, y}. A multiset corresponds to an ordinary set if the multiplicity of every element is 1. An <a href="https://en-m-wikipedia-org.translate.goog/wiki/Indexed_family?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Indexed family">indexed family</a> (ai)i∈I, where i varies over some <a href="https://en-m-wikipedia-org.translate.goog/wiki/Index_set?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Index set">index set</a> I, may define a multiset, sometimes written {ai}. In this view the underlying set of the multiset is given by the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Image_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Image (mathematics)">image</a> of the family, and the multiplicity of any element x is the number of index values i such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}=x}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> a </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> </mrow> </msub> <mo> = </mo> <mi> x </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle a_{i}=x} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b561c9515f6a90ce3f72cd5af3014b85e148977" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.458ex; height:2.009ex;" alt="{\displaystyle a_{i}=x}"> </noscript><span class="lazy-image-placeholder" style="width: 6.458ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b561c9515f6a90ce3f72cd5af3014b85e148977" data-alt="{\displaystyle a_{i}=x}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . In this article the multiplicities are considered to be finite, so that no element occurs infinitely many times in the family; even in an infinite multiset, the multiplicities are finite numbers. It is possible to extend the definition of a multiset by allowing multiplicities of individual elements to be infinite <a href="https://en-m-wikipedia-org.translate.goog/wiki/Cardinal_number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Cardinal number">cardinals</a> instead of positive integers, but not all properties carry over to this generalization. </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"> <h2 id="Basic_properties_and_operations">Basic properties and operations</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Multiset&action=edit&section=4&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Basic properties and operations" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-4 collapsible-block" id="mf-section-4"> Elements of a multiset are generally taken in a fixed set U, sometimes called a universe, which is often the set of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Natural_number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Natural number">natural numbers</a>. An element of U that does not belong to a given multiset is said to have a multiplicity 0 in this multiset. This extends the multiplicity function of the multiset to a function from U to the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> N </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbb {N} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.678ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" data-alt="{\displaystyle \mathbb {N} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  of non-negative integers. This defines a <a href="https://en-m-wikipedia-org.translate.goog/wiki/One-to-one_correspondence?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="One-to-one correspondence">one-to-one correspondence</a> between these functions and the multisets that have their elements in U. This extended multiplicity function is commonly called simply the multiplicity function, and suffices for defining multisets when the universe containing the elements has been fixed. This multiplicity function is a generalization of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Indicator_function?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Indicator function">indicator function</a> of a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Subset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Subset">subset</a>, and shares some properties with it. The support of a multiset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> A </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle A} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> </noscript><span class="lazy-image-placeholder" style="width: 1.743ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" data-alt="{\displaystyle A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  in a universe U is the underlying set of the multiset. Using the multiplicity function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> m </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> </noscript><span class="lazy-image-placeholder" style="width: 2.04ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" data-alt="{\displaystyle m}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , it is characterized as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Supp} (A):=\{x\in U\mid m_{A}(x)>0\}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> Supp </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mi> A </mi> <mo stretchy="false"> ) </mo> <mo> := </mo> <mo fence="false" stretchy="false"> { </mo> <mi> x </mi> <mo> ∈ </mo> <mi> U </mi> <mo> ∣ </mo> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> A </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> > </mo> <mn> 0 </mn> <mo fence="false" stretchy="false"> } </mo> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \operatorname {Supp} (A):=\{x\in U\mid m_{A}(x)>0\}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4de66657e1d8924701479ed8f9bfce5b08bb011b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.235ex; height:2.843ex;" alt="{\displaystyle \operatorname {Supp} (A):=\{x\in U\mid m_{A}(x)>0\}.}"> </noscript><span class="lazy-image-placeholder" style="width: 34.235ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4de66657e1d8924701479ed8f9bfce5b08bb011b" data-alt="{\displaystyle \operatorname {Supp} (A):=\{x\in U\mid m_{A}(x)>0\}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  A multiset is finite if its support is finite, or, equivalently, if its cardinality <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |A|=\sum _{x\in \operatorname {Supp} (A)}m_{A}(x)=\sum _{x\in U}m_{A}(x)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mo> = </mo> <munder> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> <mo> ∈ </mo> <mi> Supp </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mi> A </mi> <mo stretchy="false"> ) </mo> </mrow> </munder> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> A </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <munder> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> x </mi> <mo> ∈ </mo> <mi> U </mi> </mrow> </munder> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> A </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle |A|=\sum _{x\in \operatorname {Supp} (A)}m_{A}(x)=\sum _{x\in U}m_{A}(x)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0518a72400bc45e77381435ac3cb66577741a46b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:34.855ex; height:6.009ex;" alt="{\displaystyle |A|=\sum _{x\in \operatorname {Supp} (A)}m_{A}(x)=\sum _{x\in U}m_{A}(x)}"> </noscript><span class="lazy-image-placeholder" style="width: 34.855ex;height: 6.009ex;vertical-align: -3.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0518a72400bc45e77381435ac3cb66577741a46b" data-alt="{\displaystyle |A|=\sum _{x\in \operatorname {Supp} (A)}m_{A}(x)=\sum _{x\in U}m_{A}(x)}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  is finite. The empty multiset is the unique multiset with an <a href="https://en-m-wikipedia-org.translate.goog/wiki/Empty_set?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Empty set">empty</a> support (underlying set), and thus a cardinality 0. The usual operations of sets may be extended to multisets by using the multiplicity function, in a similar way to using the indicator function for subsets. In the following, A and B are multisets in a given universe U, with multiplicity functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{A}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> A </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m_{A}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b38fd8717ed74667ad31b45948df4058db8ddb35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.505ex; height:2.009ex;" alt="{\displaystyle m_{A}}"> </noscript><span class="lazy-image-placeholder" style="width: 3.505ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b38fd8717ed74667ad31b45948df4058db8ddb35" data-alt="{\displaystyle m_{A}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{B}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> B </mi> </mrow> </msub> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m_{B}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b03be0d99db960a1dcdf30c66aec3ad3dcd42b5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.167ex; height:2.009ex;" alt="{\displaystyle m_{B}.}"> </noscript><span class="lazy-image-placeholder" style="width: 4.167ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b03be0d99db960a1dcdf30c66aec3ad3dcd42b5f" data-alt="{\displaystyle m_{B}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  <ul> <li>Inclusion: A is included in B, denoted A ⊆ B, if <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{A}(x)\leq m_{B}(x)\quad \forall x\in U.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> A </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> ≤ </mo> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> B </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mspace width="1em"></mspace> <mi mathvariant="normal"> ∀ </mi> <mi> x </mi> <mo> ∈ </mo> <mi> U </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m_{A}(x)\leq m_{B}(x)\quad \forall x\in U.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed7a0f4521a8a7985f40beee4e4e8d936eb60b77" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.617ex; height:2.843ex;" alt="{\displaystyle m_{A}(x)\leq m_{B}(x)\quad \forall x\in U.}"> </noscript><span class="lazy-image-placeholder" style="width: 26.617ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed7a0f4521a8a7985f40beee4e4e8d936eb60b77" data-alt="{\displaystyle m_{A}(x)\leq m_{B}(x)\quad \forall x\in U.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> </li> <li>Union: the union (called, in some contexts, the maximum or lowest common multiple) of A and B is the multiset C with multiplicity function<a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-syropoulos-13">[13]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{C}(x)=\max(m_{A}(x),m_{B}(x))\quad \forall x\in U.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> C </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mo movablelimits="true" form="prefix"> max </mo> <mo stretchy="false"> ( </mo> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> A </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> , </mo> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> B </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mspace width="1em"></mspace> <mi mathvariant="normal"> ∀ </mi> <mi> x </mi> <mo> ∈ </mo> <mi> U </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m_{C}(x)=\max(m_{A}(x),m_{B}(x))\quad \forall x\in U.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31ac7c0a50ea1ac9b87b519b2be538c6c663358b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.446ex; height:2.843ex;" alt="{\displaystyle m_{C}(x)=\max(m_{A}(x),m_{B}(x))\quad \forall x\in U.}"> </noscript><span class="lazy-image-placeholder" style="width: 40.446ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31ac7c0a50ea1ac9b87b519b2be538c6c663358b" data-alt="{\displaystyle m_{C}(x)=\max(m_{A}(x),m_{B}(x))\quad \forall x\in U.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> </li> <li>Intersection: the intersection (called, in some contexts, the infimum or greatest common divisor) of A and B is the multiset C with multiplicity function <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{C}(x)=\min(m_{A}(x),m_{B}(x))\quad \forall x\in U.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> C </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mo movablelimits="true" form="prefix"> min </mo> <mo stretchy="false"> ( </mo> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> A </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> , </mo> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> B </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mspace width="1em"></mspace> <mi mathvariant="normal"> ∀ </mi> <mi> x </mi> <mo> ∈ </mo> <mi> U </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m_{C}(x)=\min(m_{A}(x),m_{B}(x))\quad \forall x\in U.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16dd168b1555393b7c516ba2409139e3adf2c14f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.996ex; height:2.843ex;" alt="{\displaystyle m_{C}(x)=\min(m_{A}(x),m_{B}(x))\quad \forall x\in U.}"> </noscript><span class="lazy-image-placeholder" style="width: 39.996ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16dd168b1555393b7c516ba2409139e3adf2c14f" data-alt="{\displaystyle m_{C}(x)=\min(m_{A}(x),m_{B}(x))\quad \forall x\in U.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> </li> <li>Sum: the sum of A and B is the multiset C with multiplicity function <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{C}(x)=m_{A}(x)+m_{B}(x)\quad \forall x\in U.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> C </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> A </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> + </mo> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> B </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mspace width="1em"></mspace> <mi mathvariant="normal"> ∀ </mi> <mi> x </mi> <mo> ∈ </mo> <mi> U </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m_{C}(x)=m_{A}(x)+m_{B}(x)\quad \forall x\in U.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cd040ecf0fb3f495ba58fc2093acc9aba32c52f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.118ex; height:2.843ex;" alt="{\displaystyle m_{C}(x)=m_{A}(x)+m_{B}(x)\quad \forall x\in U.}"> </noscript><span class="lazy-image-placeholder" style="width: 36.118ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cd040ecf0fb3f495ba58fc2093acc9aba32c52f" data-alt="{\displaystyle m_{C}(x)=m_{A}(x)+m_{B}(x)\quad \forall x\in U.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  It may be viewed as a generalization of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Disjoint_union?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Disjoint union">disjoint union</a> of sets. It defines a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Commutative_monoid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Commutative monoid">commutative monoid</a> structure on the finite multisets in a given universe. This monoid is a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Free_commutative_monoid?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Free commutative monoid">free commutative monoid</a>, with the universe as a basis.</li> <li>Difference: the difference of A and B is the multiset C with multiplicity function <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{C}(x)=\max(m_{A}(x)-m_{B}(x),0)\quad \forall x\in U.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> C </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> = </mo> <mo movablelimits="true" form="prefix"> max </mo> <mo stretchy="false"> ( </mo> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> A </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> − </mo> <msub> <mi> m </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> B </mi> </mrow> </msub> <mo stretchy="false"> ( </mo> <mi> x </mi> <mo stretchy="false"> ) </mo> <mo> , </mo> <mn> 0 </mn> <mo stretchy="false"> ) </mo> <mspace width="1em"></mspace> <mi mathvariant="normal"> ∀ </mi> <mi> x </mi> <mo> ∈ </mo> <mi> U </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle m_{C}(x)=\max(m_{A}(x)-m_{B}(x),0)\quad \forall x\in U.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80a917a0d6ed8f0e4dacdca7b33792d0858ce725" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.449ex; height:2.843ex;" alt="{\displaystyle m_{C}(x)=\max(m_{A}(x)-m_{B}(x),0)\quad \forall x\in U.}"> </noscript><span class="lazy-image-placeholder" style="width: 44.449ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80a917a0d6ed8f0e4dacdca7b33792d0858ce725" data-alt="{\displaystyle m_{C}(x)=\max(m_{A}(x)-m_{B}(x),0)\quad \forall x\in U.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> </li> </ul> Two multisets are disjoint if their supports are <a href="https://en-m-wikipedia-org.translate.goog/wiki/Disjoint_sets?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Disjoint sets">disjoint sets</a>. This is equivalent to saying that their intersection is the empty multiset or that their sum equals their union. There is an inclusion–exclusion principle for finite multisets (similar to <a href="https://en-m-wikipedia-org.translate.goog/wiki/Inclusion%E2%80%93exclusion_principle?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Inclusion–exclusion principle">the one for sets</a>), stating that a finite union of finite multisets is the difference of two sums of multisets: in the first sum we consider all possible intersections of an <a href="https://en-m-wikipedia-org.translate.goog/wiki/Parity_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Parity (mathematics)">odd</a> number of the given multisets, while in the second sum we consider all possible intersections of an <a href="https://en-m-wikipedia-org.translate.goog/wiki/Parity_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Parity (mathematics)">even</a> number of the given multisets.[<a href="https://en-m-wikipedia-org.translate.goog/wiki/Wikipedia:Citation_needed?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Wikipedia:Citation needed">citation needed</a>] </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"> <h2 id="Counting_multisets">Counting multisets</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Multiset&action=edit&section=5&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Counting multisets" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-5 collapsible-block" id="mf-section-5"> <figure typeof="mw:File/Thumb"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Combinations_with_repetition;_5_multichoose_3.svg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Combinations_with_repetition%3B_5_multichoose_3.svg/370px-Combinations_with_repetition%3B_5_multichoose_3.svg.png" decoding="async" width="370" height="405" class="mw-file-element" data-file-width="626" data-file-height="685"> </noscript><span class="lazy-image-placeholder" style="width: 370px;height: 405px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Combinations_with_repetition%3B_5_multichoose_3.svg/370px-Combinations_with_repetition%3B_5_multichoose_3.svg.png" data-width="370" data-height="405" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Combinations_with_repetition%3B_5_multichoose_3.svg/555px-Combinations_with_repetition%3B_5_multichoose_3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/Combinations_with_repetition%3B_5_multichoose_3.svg/740px-Combinations_with_repetition%3B_5_multichoose_3.svg.png 2x" data-class="mw-file-element"> </a> <figcaption> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Bijection?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Bijection">Bijection</a> between 3-subsets of a 7-set (left) and 3-multisets with elements from a 5-set (right) So this illustrates that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {7 \choose 3}=\left(\!\!{5 \choose 3}\!\!\right).}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em"> ( </mo> </mrow> <mfrac linethickness="0"> <mn> 7 </mn> <mn> 3 </mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em"> ) </mo> </mrow> </mrow> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em"> ( </mo> </mrow> <mfrac linethickness="0"> <mn> 5 </mn> <mn> 3 </mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em"> ) </mo> </mrow> </mrow> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mrow> <mo> ) </mo> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\textstyle {7 \choose 3}=\left(\!\!{5 \choose 3}\!\!\right).} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c12f5fc13644266bb9aa331ada0e9af3b2ff968a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.617ex; height:3.509ex;" alt="{\textstyle {7 \choose 3}=\left(\!\!{5 \choose 3}\!\!\right).}"> </noscript><span class="lazy-image-placeholder" style="width: 10.617ex;height: 3.509ex;vertical-align: -1.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c12f5fc13644266bb9aa331ada0e9af3b2ff968a" data-alt="{\textstyle {7 \choose 3}=\left(\!\!{5 \choose 3}\!\!\right).}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  </figcaption> </figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"> <div role="note" class="hatnote navigation-not-searchable"> See also: <a href="https://en-m-wikipedia-org.translate.goog/wiki/Stars_and_bars_(combinatorics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Stars and bars (combinatorics)">Stars and bars (combinatorics)</a> </div> The number of multisets of cardinality k, with elements taken from a finite set of cardinality n, is sometimes called the multiset coefficient or multiset number. This number is written by some authors as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \left(\!\!{n \choose k}\!\!\right)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mo> ( </mo> <mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em"> ( </mo> </mrow> <mfrac linethickness="0"> <mi> n </mi> <mi> k </mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em"> ) </mo> </mrow> </mrow> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mrow> <mo> ) </mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \textstyle \left(\!\!{n \choose k}\!\!\right)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/646388f9720fa95a72e8947270361909ce61e4f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.697ex; height:3.176ex;" alt="{\displaystyle \textstyle \left(\!\!{n \choose k}\!\!\right)}"> </noscript><span class="lazy-image-placeholder" style="width: 3.697ex;height: 3.176ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/646388f9720fa95a72e8947270361909ce61e4f0" data-alt="{\displaystyle \textstyle \left(\!\!{n \choose k}\!\!\right)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , a notation that is meant to resemble that of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Binomial_coefficient?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Binomial coefficient">binomial coefficients</a>; it is used for instance in (Stanley, 1997), and could be pronounced "n multichoose k" to resemble "n choose k" for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tbinom {n}{k}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em"> ( </mo> </mrow> <mfrac linethickness="0"> <mi> n </mi> <mi> k </mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em"> ) </mo> </mrow> </mrow> </mstyle> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\tbinom {n}{k}}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39c7822635a8fa426d00ca72733ea1bd6fe90b01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.763ex; height:3.176ex;" alt="{\displaystyle {\tbinom {n}{k}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 3.763ex;height: 3.176ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39c7822635a8fa426d00ca72733ea1bd6fe90b01" data-alt="{\displaystyle {\tbinom {n}{k}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  Like the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Binomial_distribution?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Binomial distribution">binomial distribution</a> that involves binomial coefficients, there is a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Negative_binomial_distribution?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Negative binomial distribution">negative binomial distribution</a> in which the multiset coefficients occur. Multiset coefficients should not be confused with the unrelated <a href="https://en-m-wikipedia-org.translate.goog/wiki/Multinomial_coefficient?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Multinomial coefficient">multinomial coefficients</a> that occur in the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Multinomial_theorem?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Multinomial theorem">multinomial theorem</a>. The value of multiset coefficients can be given explicitly as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\!\!{n \choose k}\!\!\right)={n+k-1 \choose k}={\frac {(n+k-1)!}{k!\,(n-1)!}}={n(n+1)(n+2)\cdots (n+k-1) \over k!},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo> ( </mo> <mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mi> n </mi> <mi> k </mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mrow> <mi> n </mi> <mo> + </mo> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> + </mo> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> <mspace width="thinmathspace"></mspace> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> n </mi> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> <mo stretchy="false"> ) </mo> <mo> ⋯ </mo> <mo stretchy="false"> ( </mo> <mi> n </mi> <mo> + </mo> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left(\!\!{n \choose k}\!\!\right)={n+k-1 \choose k}={\frac {(n+k-1)!}{k!\,(n-1)!}}={n(n+1)(n+2)\cdots (n+k-1) \over k!},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3085c625fdc63db0b0cf48f07516a78a87109a56" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:73.643ex; height:6.509ex;" alt="{\displaystyle \left(\!\!{n \choose k}\!\!\right)={n+k-1 \choose k}={\frac {(n+k-1)!}{k!\,(n-1)!}}={n(n+1)(n+2)\cdots (n+k-1) \over k!},}"> </noscript><span class="lazy-image-placeholder" style="width: 73.643ex;height: 6.509ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3085c625fdc63db0b0cf48f07516a78a87109a56" data-alt="{\displaystyle \left(\!\!{n \choose k}\!\!\right)={n+k-1 \choose k}={\frac {(n+k-1)!}{k!\,(n-1)!}}={n(n+1)(n+2)\cdots (n+k-1) \over k!},}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  where the second expression is as a binomial coefficient;<a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-17">[a]</a> many authors in fact avoid separate notation and just write binomial coefficients. So, the number of such multisets is the same as the number of subsets of cardinality k of a set of cardinality n + k − 1. The analogy with binomial coefficients can be stressed by writing the numerator in the above expression as a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Rising_factorial_power?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Rising factorial power">rising factorial power</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\!\!{n \choose k}\!\!\right)={n^{\overline {k}} \over k!},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo> ( </mo> <mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mi> n </mi> <mi> k </mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> </msup> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left(\!\!{n \choose k}\!\!\right)={n^{\overline {k}} \over k!},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03a76bd55fd25d9a67f3fbe52bf36804ff1969a7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.847ex; height:6.843ex;" alt="{\displaystyle \left(\!\!{n \choose k}\!\!\right)={n^{\overline {k}} \over k!},}"> </noscript><span class="lazy-image-placeholder" style="width: 13.847ex;height: 6.843ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03a76bd55fd25d9a67f3fbe52bf36804ff1969a7" data-alt="{\displaystyle \left(\!\!{n \choose k}\!\!\right)={n^{\overline {k}} \over k!},}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  to match the expression of binomial coefficients using a falling factorial power: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {n \choose k}={n^{\underline {k}} \over k!}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mi> n </mi> <mi> k </mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi> n </mi> <mrow class="MJX-TeXAtom-ORD"> <munder> <mi> k </mi> <mo> _ </mo> </munder> </mrow> </msup> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {n \choose k}={n^{\underline {k}} \over k!}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dc84d073d8ca2a395d37ff296d4ae1f04653b3e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.882ex; height:6.343ex;" alt="{\displaystyle {n \choose k}={n^{\underline {k}} \over k!}.}"> </noscript><span class="lazy-image-placeholder" style="width: 11.882ex;height: 6.343ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dc84d073d8ca2a395d37ff296d4ae1f04653b3e" data-alt="{\displaystyle {n \choose k}={n^{\underline {k}} \over k!}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  For example, there are 4 multisets of cardinality 3 with elements taken from the set {1, 2} of cardinality 2 (n = 2, k = 3), namely {1, 1, 1}, {1, 1, 2}, {1, 2, 2}, {2, 2, 2}. There are also 4 subsets of cardinality 3 in the set {1, 2, 3, 4} of cardinality 4 (n + k − 1), namely {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}. One simple way to <a href="https://en-m-wikipedia-org.translate.goog/wiki/Mathematical_proof?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mathematical proof">prove</a> the equality of multiset coefficients and binomial coefficients given above involves representing multisets in the following way. First, consider the notation for multisets that would represent {a, a, a, a, a, a, b, b, c, c, c, d, d, d, d, d, d, d} (6 as, 2 bs, 3 cs, 7 ds) in this form: <dl> <dd>  •  •  •  •  •  •  |  •  •  |  •  •  •  |  •  •  •  •  •  •  • </dd> </dl> This is a multiset of cardinality k = 18 made of elements of a set of cardinality n = 4. The number of characters including both dots and vertical lines used in this notation is 18 + 4 − 1. The number of vertical lines is 4 − 1. The number of multisets of cardinality 18 is then the number of ways to arrange the 4 − 1 vertical lines among the 18 + 4 − 1 characters, and is thus the number of subsets of cardinality 4 − 1 of a set of cardinality 18 + 4 − 1. Equivalently, it is the number of ways to arrange the 18 dots among the 18 + 4 − 1 characters, which is the number of subsets of cardinality 18 of a set of cardinality 18 + 4 − 1. This is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {4+18-1 \choose 4-1}={4+18-1 \choose 18}=1330,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mrow> <mn> 4 </mn> <mo> + </mo> <mn> 18 </mn> <mo> − </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 4 </mn> <mo> − </mo> <mn> 1 </mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mrow> <mn> 4 </mn> <mo> + </mo> <mn> 18 </mn> <mo> − </mo> <mn> 1 </mn> </mrow> <mn> 18 </mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <mo> = </mo> <mn> 1330 </mn> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {4+18-1 \choose 4-1}={4+18-1 \choose 18}=1330,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46065be8842febd05224cfbc91bee1a4e5024eb1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.997ex; height:6.176ex;" alt="{\displaystyle {4+18-1 \choose 4-1}={4+18-1 \choose 18}=1330,}"> </noscript><span class="lazy-image-placeholder" style="width: 38.997ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46065be8842febd05224cfbc91bee1a4e5024eb1" data-alt="{\displaystyle {4+18-1 \choose 4-1}={4+18-1 \choose 18}=1330,}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  thus is the value of the multiset coefficient and its equivalencies: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\left(\!\!{4 \choose 18}\!\!\right)&={21 \choose 18}={\frac {21!}{18!\,3!}}={21 \choose 3},\\[1ex]&={\frac {{\color {red}{\mathfrak {4\cdot 5\cdot 6\cdot 7\cdot 8\cdot 9\cdot 10\cdot 11\cdot 12\cdot 13\cdot 14\cdot 15\cdot 16\cdot 17\cdot 18}}}\cdot \mathbf {19\cdot 20\cdot 21} }{\mathbf {1\cdot 2\cdot 3} \cdot {\color {red}{\mathfrak {4\cdot 5\cdot 6\cdot 7\cdot 8\cdot 9\cdot 10\cdot 11\cdot 12\cdot 13\cdot 14\cdot 15\cdot 16\cdot 17\cdot 18}}}}},\\[1ex]&={\frac {1\cdot 2\cdot 3\cdot 4\cdot 5\cdots 16\cdot 17\cdot 18\;\mathbf {\cdot \;19\cdot 20\cdot 21} }{\,1\cdot 2\cdot 3\cdot 4\cdot 5\cdots 16\cdot 17\cdot 18\;\mathbf {\cdot \;1\cdot 2\cdot 3\quad } }},\\[1ex]&={\frac {19\cdot 20\cdot 21}{1\cdot 2\cdot 3}}.\end{aligned}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.73em 0.73em 0.73em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow> <mo> ( </mo> <mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mn> 4 </mn> <mn> 18 </mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mrow> <mo> ) </mo> </mrow> </mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mn> 21 </mn> <mn> 18 </mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 21 </mn> <mo> ! </mo> </mrow> <mrow> <mn> 18 </mn> <mo> ! </mo> <mspace width="thinmathspace"></mspace> <mn> 3 </mn> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mn> 21 </mn> <mn> 3 </mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <mo> , </mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="fraktur"> 4 </mn> <mo> ⋅ </mo> <mn mathvariant="fraktur"> 5 </mn> <mo> ⋅ </mo> <mn mathvariant="fraktur"> 6 </mn> <mo> ⋅ </mo> <mn mathvariant="fraktur"> 7 </mn> <mo> ⋅ </mo> <mn mathvariant="fraktur"> 8 </mn> <mo> ⋅ </mo> <mn mathvariant="fraktur"> 9 </mn> <mo> ⋅ </mo> <mn mathvariant="fraktur"> 10 </mn> <mo> ⋅ </mo> <mn mathvariant="fraktur"> 11 </mn> <mo> ⋅ </mo> <mn mathvariant="fraktur"> 12 </mn> <mo> ⋅ </mo> <mn mathvariant="fraktur"> 13 </mn> <mo> ⋅ </mo> <mn mathvariant="fraktur"> 14 </mn> <mo> ⋅ </mo> <mn mathvariant="fraktur"> 15 </mn> <mo> ⋅ </mo> <mn mathvariant="fraktur"> 16 </mn> <mo> ⋅ </mo> <mn mathvariant="fraktur"> 17 </mn> <mo> ⋅ </mo> <mn mathvariant="fraktur"> 18 </mn> </mrow> </mrow> </mstyle> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold"> 19 </mn> <mo> ⋅ </mo> <mn mathvariant="bold"> 20 </mn> <mo> ⋅ </mo> <mn mathvariant="bold"> 21 </mn> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold"> 1 </mn> <mo> ⋅ </mo> <mn mathvariant="bold"> 2 </mn> <mo> ⋅ </mo> <mn mathvariant="bold"> 3 </mn> </mrow> <mo> ⋅ </mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="red"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="fraktur"> 4 </mn> <mo> ⋅ </mo> <mn mathvariant="fraktur"> 5 </mn> <mo> ⋅ </mo> <mn mathvariant="fraktur"> 6 </mn> <mo> ⋅ </mo> <mn mathvariant="fraktur"> 7 </mn> <mo> ⋅ </mo> <mn mathvariant="fraktur"> 8 </mn> <mo> ⋅ </mo> <mn mathvariant="fraktur"> 9 </mn> <mo> ⋅ </mo> <mn mathvariant="fraktur"> 10 </mn> <mo> ⋅ </mo> <mn mathvariant="fraktur"> 11 </mn> <mo> ⋅ </mo> <mn mathvariant="fraktur"> 12 </mn> <mo> ⋅ </mo> <mn mathvariant="fraktur"> 13 </mn> <mo> ⋅ </mo> <mn mathvariant="fraktur"> 14 </mn> <mo> ⋅ </mo> <mn mathvariant="fraktur"> 15 </mn> <mo> ⋅ </mo> <mn mathvariant="fraktur"> 16 </mn> <mo> ⋅ </mo> <mn mathvariant="fraktur"> 17 </mn> <mo> ⋅ </mo> <mn mathvariant="fraktur"> 18 </mn> </mrow> </mrow> </mstyle> </mrow> </mrow> </mfrac> </mrow> <mo> , </mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 1 </mn> <mo> ⋅ </mo> <mn> 2 </mn> <mo> ⋅ </mo> <mn> 3 </mn> <mo> ⋅ </mo> <mn> 4 </mn> <mo> ⋅ </mo> <mn> 5 </mn> <mo> ⋯ </mo> <mn> 16 </mn> <mo> ⋅ </mo> <mn> 17 </mn> <mo> ⋅ </mo> <mn> 18 </mn> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mo> ⋅ </mo> <mspace width="thickmathspace"></mspace> <mn mathvariant="bold"> 19 </mn> <mo> ⋅ </mo> <mn mathvariant="bold"> 20 </mn> <mo> ⋅ </mo> <mn mathvariant="bold"> 21 </mn> </mrow> </mrow> <mrow> <mspace width="thinmathspace"></mspace> <mn> 1 </mn> <mo> ⋅ </mo> <mn> 2 </mn> <mo> ⋅ </mo> <mn> 3 </mn> <mo> ⋅ </mo> <mn> 4 </mn> <mo> ⋅ </mo> <mn> 5 </mn> <mo> ⋯ </mo> <mn> 16 </mn> <mo> ⋅ </mo> <mn> 17 </mn> <mo> ⋅ </mo> <mn> 18 </mn> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mo> ⋅ </mo> <mspace width="thickmathspace"></mspace> <mn mathvariant="bold"> 1 </mn> <mo> ⋅ </mo> <mn mathvariant="bold"> 2 </mn> <mo> ⋅ </mo> <mn mathvariant="bold"> 3 </mn> <mspace width="1em"></mspace> </mrow> </mrow> </mfrac> </mrow> <mo> , </mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn> 19 </mn> <mo> ⋅ </mo> <mn> 20 </mn> <mo> ⋅ </mo> <mn> 21 </mn> </mrow> <mrow> <mn> 1 </mn> <mo> ⋅ </mo> <mn> 2 </mn> <mo> ⋅ </mo> <mn> 3 </mn> </mrow> </mfrac> </mrow> <mo> . </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}\left(\!\!{4 \choose 18}\!\!\right)&={21 \choose 18}={\frac {21!}{18!\,3!}}={21 \choose 3},\\[1ex]&={\frac {{\color {red}{\mathfrak {4\cdot 5\cdot 6\cdot 7\cdot 8\cdot 9\cdot 10\cdot 11\cdot 12\cdot 13\cdot 14\cdot 15\cdot 16\cdot 17\cdot 18}}}\cdot \mathbf {19\cdot 20\cdot 21} }{\mathbf {1\cdot 2\cdot 3} \cdot {\color {red}{\mathfrak {4\cdot 5\cdot 6\cdot 7\cdot 8\cdot 9\cdot 10\cdot 11\cdot 12\cdot 13\cdot 14\cdot 15\cdot 16\cdot 17\cdot 18}}}}},\\[1ex]&={\frac {1\cdot 2\cdot 3\cdot 4\cdot 5\cdots 16\cdot 17\cdot 18\;\mathbf {\cdot \;19\cdot 20\cdot 21} }{\,1\cdot 2\cdot 3\cdot 4\cdot 5\cdots 16\cdot 17\cdot 18\;\mathbf {\cdot \;1\cdot 2\cdot 3\quad } }},\\[1ex]&={\frac {19\cdot 20\cdot 21}{1\cdot 2\cdot 3}}.\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37ec5dd66fc50e14645949792c2330cddb34e6a4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.505ex; width:77.527ex; height:26.176ex;" alt="{\displaystyle {\begin{aligned}\left(\!\!{4 \choose 18}\!\!\right)&={21 \choose 18}={\frac {21!}{18!\,3!}}={21 \choose 3},\\[1ex]&={\frac {{\color {red}{\mathfrak {4\cdot 5\cdot 6\cdot 7\cdot 8\cdot 9\cdot 10\cdot 11\cdot 12\cdot 13\cdot 14\cdot 15\cdot 16\cdot 17\cdot 18}}}\cdot \mathbf {19\cdot 20\cdot 21} }{\mathbf {1\cdot 2\cdot 3} \cdot {\color {red}{\mathfrak {4\cdot 5\cdot 6\cdot 7\cdot 8\cdot 9\cdot 10\cdot 11\cdot 12\cdot 13\cdot 14\cdot 15\cdot 16\cdot 17\cdot 18}}}}},\\[1ex]&={\frac {1\cdot 2\cdot 3\cdot 4\cdot 5\cdots 16\cdot 17\cdot 18\;\mathbf {\cdot \;19\cdot 20\cdot 21} }{\,1\cdot 2\cdot 3\cdot 4\cdot 5\cdots 16\cdot 17\cdot 18\;\mathbf {\cdot \;1\cdot 2\cdot 3\quad } }},\\[1ex]&={\frac {19\cdot 20\cdot 21}{1\cdot 2\cdot 3}}.\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 77.527ex;height: 26.176ex;vertical-align: -12.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37ec5dd66fc50e14645949792c2330cddb34e6a4" data-alt="{\displaystyle {\begin{aligned}\left(\!\!{4 \choose 18}\!\!\right)&={21 \choose 18}={\frac {21!}{18!\,3!}}={21 \choose 3},\\[1ex]&={\frac {{\color {red}{\mathfrak {4\cdot 5\cdot 6\cdot 7\cdot 8\cdot 9\cdot 10\cdot 11\cdot 12\cdot 13\cdot 14\cdot 15\cdot 16\cdot 17\cdot 18}}}\cdot \mathbf {19\cdot 20\cdot 21} }{\mathbf {1\cdot 2\cdot 3} \cdot {\color {red}{\mathfrak {4\cdot 5\cdot 6\cdot 7\cdot 8\cdot 9\cdot 10\cdot 11\cdot 12\cdot 13\cdot 14\cdot 15\cdot 16\cdot 17\cdot 18}}}}},\\[1ex]&={\frac {1\cdot 2\cdot 3\cdot 4\cdot 5\cdots 16\cdot 17\cdot 18\;\mathbf {\cdot \;19\cdot 20\cdot 21} }{\,1\cdot 2\cdot 3\cdot 4\cdot 5\cdots 16\cdot 17\cdot 18\;\mathbf {\cdot \;1\cdot 2\cdot 3\quad } }},\\[1ex]&={\frac {19\cdot 20\cdot 21}{1\cdot 2\cdot 3}}.\end{aligned}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  From the relation between binomial coefficients and multiset coefficients, it follows that the number of multisets of cardinality k in a set of cardinality n can be written <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\!\!{n \choose k}\!\!\right)=(-1)^{k}{-n \choose k}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo> ( </mo> <mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mi> n </mi> <mi> k </mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mo stretchy="false"> ( </mo> <mo> − </mo> <mn> 1 </mn> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mrow> <mo> − </mo> <mi> n </mi> </mrow> <mi> k </mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left(\!\!{n \choose k}\!\!\right)=(-1)^{k}{-n \choose k}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/686bb689a4e560b2fbf2edbb867fbbf411670bf3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.927ex; height:6.176ex;" alt="{\displaystyle \left(\!\!{n \choose k}\!\!\right)=(-1)^{k}{-n \choose k}.}"> </noscript><span class="lazy-image-placeholder" style="width: 22.927ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/686bb689a4e560b2fbf2edbb867fbbf411670bf3" data-alt="{\displaystyle \left(\!\!{n \choose k}\!\!\right)=(-1)^{k}{-n \choose k}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  Additionally, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\!\!{n \choose k}\!\!\right)=\left(\!\!{k+1 \choose n-1}\!\!\right).}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo> ( </mo> <mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mi> n </mi> <mi> k </mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mrow> <mo> ) </mo> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left(\!\!{n \choose k}\!\!\right)=\left(\!\!{k+1 \choose n-1}\!\!\right).} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/109d5fda4a74e69fec93c090cda923082e7fdca1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.513ex; height:6.176ex;" alt="{\displaystyle \left(\!\!{n \choose k}\!\!\right)=\left(\!\!{k+1 \choose n-1}\!\!\right).}"> </noscript><span class="lazy-image-placeholder" style="width: 21.513ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/109d5fda4a74e69fec93c090cda923082e7fdca1" data-alt="{\displaystyle \left(\!\!{n \choose k}\!\!\right)=\left(\!\!{k+1 \choose n-1}\!\!\right).}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  <div class="mw-heading mw-heading3"> <h3 id="Recurrence_relation">Recurrence relation</h3> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Multiset&action=edit&section=6&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Recurrence relation" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> A <a href="https://en-m-wikipedia-org.translate.goog/wiki/Recurrence_relation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Recurrence relation">recurrence relation</a> for multiset coefficients may be given as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\!\!{n \choose k}\!\!\right)=\left(\!\!{n \choose k-1}\!\!\right)+\left(\!\!{n-1 \choose k}\!\!\right)\quad {\mbox{for }}n,k>0}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo> ( </mo> <mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mi> n </mi> <mi> k </mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mi> n </mi> <mrow> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mo> ( </mo> <mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mrow> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mrow> <mo> ) </mo> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> for  </mtext> </mstyle> </mrow> <mi> n </mi> <mo> , </mo> <mi> k </mi> <mo> > </mo> <mn> 0 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left(\!\!{n \choose k}\!\!\right)=\left(\!\!{n \choose k-1}\!\!\right)+\left(\!\!{n-1 \choose k}\!\!\right)\quad {\mbox{for }}n,k>0} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/845ce7b53eff000466674250587d4f11bdec7ac3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.804ex; height:6.176ex;" alt="{\displaystyle \left(\!\!{n \choose k}\!\!\right)=\left(\!\!{n \choose k-1}\!\!\right)+\left(\!\!{n-1 \choose k}\!\!\right)\quad {\mbox{for }}n,k>0}"> </noscript><span class="lazy-image-placeholder" style="width: 47.804ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/845ce7b53eff000466674250587d4f11bdec7ac3" data-alt="{\displaystyle \left(\!\!{n \choose k}\!\!\right)=\left(\!\!{n \choose k-1}\!\!\right)+\left(\!\!{n-1 \choose k}\!\!\right)\quad {\mbox{for }}n,k>0}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  with <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\!\!{n \choose 0}\!\!\right)=1,\quad n\in \mathbb {N} ,\quad {\mbox{and}}\quad \left(\!\!{0 \choose k}\!\!\right)=0,\quad k>0.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo> ( </mo> <mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mi> n </mi> <mn> 0 </mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mn> 1 </mn> <mo> , </mo> <mspace width="1em"></mspace> <mi> n </mi> <mo> ∈ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> N </mi> </mrow> <mo> , </mo> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> and </mtext> </mstyle> </mrow> <mspace width="1em"></mspace> <mrow> <mo> ( </mo> <mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mn> 0 </mn> <mi> k </mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mn> 0 </mn> <mo> , </mo> <mspace width="1em"></mspace> <mi> k </mi> <mo> > </mo> <mn> 0. </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left(\!\!{n \choose 0}\!\!\right)=1,\quad n\in \mathbb {N} ,\quad {\mbox{and}}\quad \left(\!\!{0 \choose k}\!\!\right)=0,\quad k>0.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8ee422e69c1366d3e290015afcb3e14057f0fa7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:50.275ex; height:6.176ex;" alt="{\displaystyle \left(\!\!{n \choose 0}\!\!\right)=1,\quad n\in \mathbb {N} ,\quad {\mbox{and}}\quad \left(\!\!{0 \choose k}\!\!\right)=0,\quad k>0.}"> </noscript><span class="lazy-image-placeholder" style="width: 50.275ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8ee422e69c1366d3e290015afcb3e14057f0fa7" data-alt="{\displaystyle \left(\!\!{n \choose 0}\!\!\right)=1,\quad n\in \mathbb {N} ,\quad {\mbox{and}}\quad \left(\!\!{0 \choose k}\!\!\right)=0,\quad k>0.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  The above recurrence may be interpreted as follows. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [n]:=\{1,\dots ,n\}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> [ </mo> <mi> n </mi> <mo stretchy="false"> ] </mo> <mo> := </mo> <mo fence="false" stretchy="false"> { </mo> <mn> 1 </mn> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mi> n </mi> <mo fence="false" stretchy="false"> } </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle [n]:=\{1,\dots ,n\}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bef8db9982c85930794abcf92517522123bd0407" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.494ex; height:2.843ex;" alt="{\displaystyle [n]:=\{1,\dots ,n\}}"> </noscript><span class="lazy-image-placeholder" style="width: 16.494ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bef8db9982c85930794abcf92517522123bd0407" data-alt="{\displaystyle [n]:=\{1,\dots ,n\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  be the source set. There is always exactly one (empty) multiset of size 0, and if n = 0 there are no larger multisets, which gives the initial conditions. Now, consider the case in which n, k > 0. A multiset of cardinality k with elements from [n] might or might not contain any instance of the final element n. If it does appear, then by removing n once, one is left with a multiset of cardinality k − 1 of elements from [n], and every such multiset can arise, which gives a total of <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\!\!{n \choose k-1}\!\!\right)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo> ( </mo> <mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mi> n </mi> <mrow> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mrow> <mo> ) </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left(\!\!{n \choose k-1}\!\!\right)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49373df496fe6f0e08eb7569578019dde28658df" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.508ex; height:6.176ex;" alt="{\displaystyle \left(\!\!{n \choose k-1}\!\!\right)}"> </noscript><span class="lazy-image-placeholder" style="width: 10.508ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49373df496fe6f0e08eb7569578019dde28658df" data-alt="{\displaystyle \left(\!\!{n \choose k-1}\!\!\right)}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  possibilities. If n does not appear, then our original multiset is equal to a multiset of cardinality k with elements from [n − 1], of which there are <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\!\!{n-1 \choose k}\!\!\right).}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo> ( </mo> <mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mrow> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mrow> <mo> ) </mo> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left(\!\!{n-1 \choose k}\!\!\right).} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59cb57a373bd07688baa98077052fc518fbedff1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.725ex; height:6.176ex;" alt="{\displaystyle \left(\!\!{n-1 \choose k}\!\!\right).}"> </noscript><span class="lazy-image-placeholder" style="width: 11.725ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59cb57a373bd07688baa98077052fc518fbedff1" data-alt="{\displaystyle \left(\!\!{n-1 \choose k}\!\!\right).}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  Thus, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\!\!{n \choose k}\!\!\right)=\left(\!\!{n \choose k-1}\!\!\right)+\left(\!\!{n-1 \choose k}\!\!\right).}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo> ( </mo> <mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mi> n </mi> <mi> k </mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mi> n </mi> <mrow> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mo> ( </mo> <mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mrow> <mi> n </mi> <mo> − </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mrow> <mo> ) </mo> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left(\!\!{n \choose k}\!\!\right)=\left(\!\!{n \choose k-1}\!\!\right)+\left(\!\!{n-1 \choose k}\!\!\right).} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dc6c3a6ecfdeb28483542d14c62c05abc393653" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.861ex; height:6.176ex;" alt="{\displaystyle \left(\!\!{n \choose k}\!\!\right)=\left(\!\!{n \choose k-1}\!\!\right)+\left(\!\!{n-1 \choose k}\!\!\right).}"> </noscript><span class="lazy-image-placeholder" style="width: 34.861ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dc6c3a6ecfdeb28483542d14c62c05abc393653" data-alt="{\displaystyle \left(\!\!{n \choose k}\!\!\right)=\left(\!\!{n \choose k-1}\!\!\right)+\left(\!\!{n-1 \choose k}\!\!\right).}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  <div class="mw-heading mw-heading3"> <h3 id="Generating_series">Generating series</h3> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Multiset&action=edit&section=7&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Generating series" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> The <a href="https://en-m-wikipedia-org.translate.goog/wiki/Generating_function?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Generating function">generating function</a> of the multiset coefficients is very simple, being <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{d=0}^{\infty }\left(\!\!{n \choose d}\!\!\right)t^{d}={\frac {1}{(1-t)^{n}}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> d </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∞ </mi> </mrow> </munderover> <mrow> <mo> ( </mo> <mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mi> n </mi> <mi> d </mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mrow> <mo> ) </mo> </mrow> <msup> <mi> t </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> d </mi> </mrow> </msup> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <mrow> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo> − </mo> <mi> t </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \sum _{d=0}^{\infty }\left(\!\!{n \choose d}\!\!\right)t^{d}={\frac {1}{(1-t)^{n}}}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e51cd6e55b75aaa845dc16e233459f822a13ecd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:25.201ex; height:7.009ex;" alt="{\displaystyle \sum _{d=0}^{\infty }\left(\!\!{n \choose d}\!\!\right)t^{d}={\frac {1}{(1-t)^{n}}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 25.201ex;height: 7.009ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e51cd6e55b75aaa845dc16e233459f822a13ecd" data-alt="{\displaystyle \sum _{d=0}^{\infty }\left(\!\!{n \choose d}\!\!\right)t^{d}={\frac {1}{(1-t)^{n}}}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  As multisets are in one-to-one correspondence with <a href="https://en-m-wikipedia-org.translate.goog/wiki/Monomial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Monomial">monomials</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\!\!{n \choose d}\!\!\right)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo> ( </mo> <mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mi> n </mi> <mi> d </mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mrow> <mo> ) </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left(\!\!{n \choose d}\!\!\right)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/201fb6444da34b269f5fa3f2774ea61f670e60f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:6.689ex; height:6.176ex;" alt="{\displaystyle \left(\!\!{n \choose d}\!\!\right)}"> </noscript><span class="lazy-image-placeholder" style="width: 6.689ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/201fb6444da34b269f5fa3f2774ea61f670e60f9" data-alt="{\displaystyle \left(\!\!{n \choose d}\!\!\right)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is also the number of monomials of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Degree_of_a_polynomial?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Extension_to_polynomials_with_two_or_more_variables" title="Degree of a polynomial">degree</a> d in n indeterminates. Thus, the above series is also the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Hilbert_series?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Hilbert series">Hilbert series</a> of the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Polynomial_ring?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Polynomial ring">polynomial ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k[x_{1},\ldots ,x_{n}].}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> k </mi> <mo stretchy="false"> [ </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> x </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> <mo stretchy="false"> ] </mo> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle k[x_{1},\ldots ,x_{n}].} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c109f59bd548c669a7645880f875a11422061bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.262ex; height:2.843ex;" alt="{\displaystyle k[x_{1},\ldots ,x_{n}].}"> </noscript><span class="lazy-image-placeholder" style="width: 13.262ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c109f59bd548c669a7645880f875a11422061bb" data-alt="{\displaystyle k[x_{1},\ldots ,x_{n}].}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  As <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\!\!{n \choose d}\!\!\right)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo> ( </mo> <mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mi> n </mi> <mi> d </mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mrow> <mo> ) </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left(\!\!{n \choose d}\!\!\right)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/201fb6444da34b269f5fa3f2774ea61f670e60f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:6.689ex; height:6.176ex;" alt="{\displaystyle \left(\!\!{n \choose d}\!\!\right)}"> </noscript><span class="lazy-image-placeholder" style="width: 6.689ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/201fb6444da34b269f5fa3f2774ea61f670e60f9" data-alt="{\displaystyle \left(\!\!{n \choose d}\!\!\right)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is a polynomial in n, it and the generating function are well defined for any <a href="https://en-m-wikipedia-org.translate.goog/wiki/Complex_number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Complex number">complex</a> value of n. <div class="mw-heading mw-heading3"> <h3 id="Generalization_and_connection_to_the_negative_binomial_series">Generalization and connection to the negative binomial series</h3> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Multiset&action=edit&section=8&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Generalization and connection to the negative binomial series" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> The multiplicative formula allows the definition of multiset coefficients to be extended by replacing n by an arbitrary number α (negative, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Real_number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Real number">real</a>, or complex): <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\!\!{\alpha \choose k}\!\!\right)={\frac {\alpha ^{\overline {k}}}{k!}}={\frac {\alpha (\alpha +1)(\alpha +2)\cdots (\alpha +k-1)}{k(k-1)(k-2)\cdots 1}}\quad {\text{for }}k\in \mathbb {N} {\text{ and arbitrary }}\alpha .}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo> ( </mo> <mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mi> α </mi> <mi> k </mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mrow> <mo> ) </mo> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi> α </mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi> k </mi> <mo accent="false"> ¯ </mo> </mover> </mrow> </msup> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi> α </mi> <mo stretchy="false"> ( </mo> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mi> α </mi> <mo> + </mo> <mn> 2 </mn> <mo stretchy="false"> ) </mo> <mo> ⋯ </mo> <mo stretchy="false"> ( </mo> <mi> α </mi> <mo> + </mo> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mrow> <mrow> <mi> k </mi> <mo stretchy="false"> ( </mo> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ( </mo> <mi> k </mi> <mo> − </mo> <mn> 2 </mn> <mo stretchy="false"> ) </mo> <mo> ⋯ </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext> for  </mtext> </mrow> <mi> k </mi> <mo> ∈ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> N </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>  and arbitrary  </mtext> </mrow> <mi> α </mi> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left(\!\!{\alpha \choose k}\!\!\right)={\frac {\alpha ^{\overline {k}}}{k!}}={\frac {\alpha (\alpha +1)(\alpha +2)\cdots (\alpha +k-1)}{k(k-1)(k-2)\cdots 1}}\quad {\text{for }}k\in \mathbb {N} {\text{ and arbitrary }}\alpha .} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5659f439a2da55d00f4fbb0233223c2723236aa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:76.431ex; height:7.009ex;" alt="{\displaystyle \left(\!\!{\alpha \choose k}\!\!\right)={\frac {\alpha ^{\overline {k}}}{k!}}={\frac {\alpha (\alpha +1)(\alpha +2)\cdots (\alpha +k-1)}{k(k-1)(k-2)\cdots 1}}\quad {\text{for }}k\in \mathbb {N} {\text{ and arbitrary }}\alpha .}"> </noscript><span class="lazy-image-placeholder" style="width: 76.431ex;height: 7.009ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5659f439a2da55d00f4fbb0233223c2723236aa" data-alt="{\displaystyle \left(\!\!{\alpha \choose k}\!\!\right)={\frac {\alpha ^{\overline {k}}}{k!}}={\frac {\alpha (\alpha +1)(\alpha +2)\cdots (\alpha +k-1)}{k(k-1)(k-2)\cdots 1}}\quad {\text{for }}k\in \mathbb {N} {\text{ and arbitrary }}\alpha .}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  With this definition one has a generalization of the negative binomial formula (with one of the variables set to 1), which justifies calling the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\!\!{\alpha \choose k}\!\!\right)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo> ( </mo> <mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mi> α </mi> <mi> k </mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mrow> <mo> ) </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left(\!\!{\alpha \choose k}\!\!\right)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc5e1de79dde0bf8c8fd15d3ec38f369e7ec259f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:6.782ex; height:6.176ex;" alt="{\displaystyle \left(\!\!{\alpha \choose k}\!\!\right)}"> </noscript><span class="lazy-image-placeholder" style="width: 6.782ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc5e1de79dde0bf8c8fd15d3ec38f369e7ec259f" data-alt="{\displaystyle \left(\!\!{\alpha \choose k}\!\!\right)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  negative binomial coefficients: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-X)^{-\alpha }=\sum _{k=0}^{\infty }\left(\!\!{\alpha \choose k}\!\!\right)X^{k}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo> − </mo> <mi> X </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mi> α </mi> </mrow> </msup> <mo> = </mo> <munderover> <mo> ∑ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> ∞ </mi> </mrow> </munderover> <mrow> <mo> ( </mo> <mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em"> ( </mo> </mrow> <mfrac linethickness="0"> <mi> α </mi> <mi> k </mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em"> ) </mo> </mrow> </mrow> </mrow> <mspace width="negativethinmathspace"></mspace> <mspace width="negativethinmathspace"></mspace> </mrow> <mo> ) </mo> </mrow> <msup> <mi> X </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (1-X)^{-\alpha }=\sum _{k=0}^{\infty }\left(\!\!{\alpha \choose k}\!\!\right)X^{k}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bea979dd6ed915ea38172742ce4348126e93312b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.096ex; height:7.009ex;" alt="{\displaystyle (1-X)^{-\alpha }=\sum _{k=0}^{\infty }\left(\!\!{\alpha \choose k}\!\!\right)X^{k}.}"> </noscript><span class="lazy-image-placeholder" style="width: 28.096ex;height: 7.009ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bea979dd6ed915ea38172742ce4348126e93312b" data-alt="{\displaystyle (1-X)^{-\alpha }=\sum _{k=0}^{\infty }\left(\!\!{\alpha \choose k}\!\!\right)X^{k}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  This <a href="https://en-m-wikipedia-org.translate.goog/wiki/Taylor_series?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Taylor series">Taylor series</a> formula is valid for all complex numbers α and X with |X| < 1. It can also be interpreted as an <a href="https://en-m-wikipedia-org.translate.goog/wiki/Identity_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Identity (mathematics)">identity</a> of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Formal_power_series?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Formal power series">formal power series</a> in X, where it actually can serve as definition of arbitrary powers of series with constant coefficient equal to 1; the point is that with this definition all identities hold that one expects for <a href="https://en-m-wikipedia-org.translate.goog/wiki/Exponentiation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Exponentiation">exponentiation</a>, notably <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-X)^{-\alpha }(1-X)^{-\beta }=(1-X)^{-(\alpha +\beta )}\quad {\text{and}}\quad ((1-X)^{-\alpha })^{-\beta }=(1-X)^{-(-\alpha \beta )},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo> − </mo> <mi> X </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mi> α </mi> </mrow> </msup> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo> − </mo> <mi> X </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mi> β </mi> </mrow> </msup> <mo> = </mo> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo> − </mo> <mi> X </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mo stretchy="false"> ( </mo> <mi> α </mi> <mo> + </mo> <mi> β </mi> <mo stretchy="false"> ) </mo> </mrow> </msup> <mspace width="1em"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mspace width="1em"></mspace> <mo stretchy="false"> ( </mo> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo> − </mo> <mi> X </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mi> α </mi> </mrow> </msup> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mi> β </mi> </mrow> </msup> <mo> = </mo> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo> − </mo> <mi> X </mi> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mo stretchy="false"> ( </mo> <mo> − </mo> <mi> α </mi> <mi> β </mi> <mo stretchy="false"> ) </mo> </mrow> </msup> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (1-X)^{-\alpha }(1-X)^{-\beta }=(1-X)^{-(\alpha +\beta )}\quad {\text{and}}\quad ((1-X)^{-\alpha })^{-\beta }=(1-X)^{-(-\alpha \beta )},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1879548421f48121103b551c34fc6372655c480" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:78.162ex; height:3.343ex;" alt="{\displaystyle (1-X)^{-\alpha }(1-X)^{-\beta }=(1-X)^{-(\alpha +\beta )}\quad {\text{and}}\quad ((1-X)^{-\alpha })^{-\beta }=(1-X)^{-(-\alpha \beta )},}"> </noscript><span class="lazy-image-placeholder" style="width: 78.162ex;height: 3.343ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1879548421f48121103b551c34fc6372655c480" data-alt="{\displaystyle (1-X)^{-\alpha }(1-X)^{-\beta }=(1-X)^{-(\alpha +\beta )}\quad {\text{and}}\quad ((1-X)^{-\alpha })^{-\beta }=(1-X)^{-(-\alpha \beta )},}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  and formulas such as these can be used to prove identities for the multiset coefficients. If α is a nonpositive integer n, then all terms with k > −n are zero, and the infinite series becomes a finite sum. However, for other values of α, including positive integers and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Rational_number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Rational number">rational numbers</a>, the series is infinite. </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"> <h2 id="Applications">Applications</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Multiset&action=edit&section=9&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Applications" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-6 collapsible-block" id="mf-section-6"> Multisets have various applications.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Singh2007-7">[7]</a> They are becoming fundamental in <a href="https://en-m-wikipedia-org.translate.goog/wiki/Combinatorics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Combinatorics">combinatorics</a>.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Aigner1979-18">[17]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Anderson1987-19">[18]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Stanley1997-20">[19]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Stanley1999-21">[20]</a> Multisets have become an important tool in the theory of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Relational_database?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Relational database">relational databases</a>, which often uses the synonym bag.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-GrumbachMilo1996-22">[21]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-LibkinWong1994-23">[22]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-LibkingWong1995-24">[23]</a> For instance, multisets are often used to implement relations in database systems. In particular, a table (without a primary key) works as a multiset, because it can have multiple identical records. Similarly, <a href="https://en-m-wikipedia-org.translate.goog/wiki/SQL?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="SQL">SQL</a> operates on multisets and returns identical records. For instance, consider "SELECT name from Student". In the case that there are multiple records with name "Sara" in the student table, all of them are shown. That means the result of an SQL query is a multiset; if the result were instead a set, the repetitive records in the result set would have been eliminated. Another application of multisets is in modeling <a href="https://en-m-wikipedia-org.translate.goog/wiki/Multigraph?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Multigraph">multigraphs</a>. In multigraphs there can be multiple edges between any two given <a href="https://en-m-wikipedia-org.translate.goog/wiki/Vertex_(graph_theory)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Vertex (graph theory)">vertices</a>. As such, the entity that specifies the edges is a multiset, and not a set. There are also other applications. For instance, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Richard_Rado?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Richard Rado">Richard Rado</a> used multisets as a device to investigate the properties of families of sets. He wrote, "The notion of a set takes no account of multiple occurrence of any one of its members, and yet it is just this kind of information that is frequently of importance. We need only think of the set of roots of a polynomial f (x) or the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Spectrum_of_an_operator?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Spectrum of an operator">spectrum</a> of a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Linear_operator?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Linear operator">linear operator</a>."<a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Blizard1991-5">[5]</a>: 328–329 </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"> <h2 id="Generalizations">Generalizations</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Multiset&action=edit&section=10&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Generalizations" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-7 collapsible-block" id="mf-section-7"> Different generalizations of multisets have been introduced, studied and applied to solving problems. <ul> <li>Real-valued multisets (in which multiplicity of an element can be any <a href="https://en-m-wikipedia-org.translate.goog/wiki/Real_number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Real number">real number</a>)<a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-25">[24]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Blizard1990-26">[25]</a></li> <li>Fuzzy multisets<a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Yager1986-27">[26]</a></li> <li>Rough multisets<a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Grzymala-Busse1987-28">[27]</a></li> <li>Hybrid sets<a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Loeb1992-29">[28]</a></li> <li>Multisets whose multiplicity is any real-valued <a href="https://en-m-wikipedia-org.translate.goog/wiki/Step_function?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Step function">step function</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Miyamoto2001-30">[29]</a></li> <li>Soft multisets<a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Alkhazaleh2011-31">[30]</a></li> <li>Soft fuzzy multisets<a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-Alkhazaleh2012-32">[31]</a></li> <li>Named sets (unification of all generalizations of sets)<a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-33">[32]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-34">[33]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-35">[34]</a><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-36">[35]</a></li> </ul> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(8)"> <h2 id="See_also">See also</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Multiset&action=edit&section=11&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: See also" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-8 collapsible-block" id="mf-section-8"> <ul> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Frequency_(statistics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Frequency (statistics)">Frequency (statistics)</a> as multiplicity analog</li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Quasi-set_theory?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Quasi-set theory">Quasi-sets</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Set_theory?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Set theory">Set theory</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Bag-of-words_model?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Bag-of-words model">Bag-of-words model</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Wikiversity_logo_2017.svg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/16px-Wikiversity_logo_2017.svg.png" decoding="async" width="16" height="13" class="mw-file-element" data-file-width="626" data-file-height="512"> </noscript><span class="lazy-image-placeholder" style="width: 16px;height: 13px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/16px-Wikiversity_logo_2017.svg.png" data-alt="" data-width="16" data-height="13" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/24px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/32px-Wikiversity_logo_2017.svg.png 2x" data-class="mw-file-element"> </a> Learning materials related to <a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://en.wikiversity.org/wiki/Partitions_of_multisets" class="extiw" title="v:Partitions of multisets">Partitions of multisets</a> at Wikiversity</li> </ul> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(9)"> <h2 id="Notes">Notes</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Multiset&action=edit&section=12&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Notes" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-9 collapsible-block" id="mf-section-9"> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style> <div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"> <ol class="references"> <li id="cite_note-17"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-17">^</a> The formula (<style data-mw-deduplicate="TemplateStyles:r1249493202">.mw-parser-output .RMbox{box-shadow:0 2px 2px 0 rgba(0,0,0,.14),0 1px 5px 0 rgba(0,0,0,.12),0 3px 1px -2px 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table.routemap .RMr4{padding:0 0 0 3px!important;text-align:right}.mw-parser-output table.routemap>tbody>tr{line-height:1}.mw-parser-output table.routemap>tbody>tr>td,.mw-parser-output table.RMcollapse>tbody>tr>td,.mw-parser-output table.RMreplace>tbody>tr>td{padding:0;width:auto;vertical-align:middle;text-align:center}.mw-parser-output .RMir>div{display:inline-block;vertical-align:middle;padding:0;height:20px;min-height:20px}.mw-parser-output .RMir img{height:initial!important;max-width:initial!important}.mw-parser-output .RMir .RMov{position:relative}.mw-parser-output .RMir .RMov .RMic,.mw-parser-output .RMir .RMov .RMtx{position:absolute;left:0;top:0;padding:0}.mw-parser-output .RMir .RMtx{line-height:20px;height:20px;min-height:20px;vertical-align:middle;text-align:center}.mw-parser-output .RMir .RMsp{height:20px;min-height:20px}.mw-parser-output .RMir .RMtx>abbr,.mw-parser-output .RMir .RMtx>div{line-height:.975;display:inline-block;vertical-align:middle}.mw-parser-output .RMir 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.RM_cd{width:35px;min-width:35px}.mw-parser-output .RMir .RM_ocd{width:37.5px;min-width:37.5px}.mw-parser-output .RMir .RMb{width:40px;min-width:40px}.mw-parser-output .RMir .RMcb{width:45px;min-width:45px}.mw-parser-output .RMir .RMdb{width:50px;min-width:50px}.mw-parser-output .RMir .RMcdb{width:55px;min-width:55px}.mw-parser-output .RMir .RM_b{width:60px;min-width:60px}.mw-parser-output .RMir .RM_cb{width:65px;min-width:65px}.mw-parser-output .RMir .RM_db{width:70px;min-width:70px}.mw-parser-output .RMir .RM_cdb{width:75px;min-width:75px}.mw-parser-output .RMir .RMs{width:80px;min-width:80px}.mw-parser-output .RMir .RMds{width:90px;min-width:90px}.mw-parser-output .RMir .RM_s{width:100px;min-width:100px}.mw-parser-output .RMir .RM_ds{width:110px;min-width:110px}.mw-parser-output .RMir .RMbs{width:120px;min-width:120px}.mw-parser-output .RMir .RMdbs{width:130px;min-width:130px}.mw-parser-output .RMir .RM_bs{width:140px;min-width:140px}.mw-parser-output .RMir .RM_dbs{width:150px;min-width:150px}.mw-parser-output .RMir .RMw{width:160px;min-width:160px}.mw-parser-output .RMir .RM_w{width:180px;min-width:180px}.mw-parser-output .RMir .RMbw{width:200px;min-width:200px}.mw-parser-output .RMir .RM_bw{width:220px;min-width:220px}.mw-parser-output .RMir .RMsw{width:240px;min-width:240px}.mw-parser-output .RMir .RM_sw{width:260px;min-width:260px}.mw-parser-output .RMir .RMbsw{width:280px;min-width:280px}.mw-parser-output .RMir .RM_bsw{width:300px;min-width:300px}.mw-parser-output .RMsplit{font-weight:inherit;color:black;background:transparent;margin-top:-3px;margin-bottom:-3px;width:initial!important;box-sizing:initial;display:inline-table;vertical-align:middle}.mw-parser-output table.routemap .RMl>.RMsplit,.mw-parser-output table.routemap .RMr>.RMsplit{font-size:90%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .RMbox{background:inherit!important;color:white}html.skin-theme-clientpref-night .mw-parser-output .RMbox img{filter:contrast(0.4)}html.skin-theme-clientpref-night .mw-parser-output .navbar-brackets a abbr{color:white!important}html.skin-theme-clientpref-night .mw-parser-output .RMbox th{background:inherit!important;color:inherit!important;border-bottom:solid 1px #be2d2c}html.skin-theme-clientpref-night .mw-parser-output .RMbox small{color:white}html.skin-theme-clientpref-night .mw-parser-output .RMsplit{color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .RMbox img{filter:contrast(0.4)}html.skin-theme-clientpref-os .mw-parser-output .navbar-brackets a abbr{color:white!important}html.skin-theme-clientpref-os .mw-parser-output .RMbox{background:inherit!important;color:white}html.skin-theme-clientpref-os .mw-parser-output .RMbox th{background:inherit!important;color:inherit!important;border-bottom:solid 1px #be2d2c}html.skin-theme-clientpref-os .mw-parser-output .RMbox small{color:white}html.skin-theme-clientpref-os .mw-parser-output .RMsplit{color:white}}</style> <table cellspacing="0" cellpadding="0" class="RMsplit" style="text-align:center;"> <tbody> <tr> <td style="text-align:inherit;padding:0;line-height:.9;font-size:inherit">n+k −1</td> </tr> <tr> <td style="text-align:inherit;padding:0;line-height:.9;font-size:inherit">k</td> </tr> </tbody> </table> ) does not work for n = 0 (where necessarily also k = 0) if viewed as an ordinary binomial coefficient since it evaluates to ( <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1249493202"> <table cellspacing="0" cellpadding="0" class="RMsplit" style="text-align:center;"> <tbody> <tr> <td style="text-align:inherit;padding:0;line-height:.9;font-size:inherit">−1</td> </tr> <tr> <td style="text-align:inherit;padding:0;line-height:.9;font-size:inherit">0</td> </tr> </tbody> </table> ), however the formula n(n+1)(n+2)...(n+k −1)/k! does work in this case because the numerator is an <a href="https://en-m-wikipedia-org.translate.goog/wiki/Empty_product?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Empty product">empty product</a> giving 1/0! = 1. However ( <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1249493202"> <table cellspacing="0" cellpadding="0" class="RMsplit" style="text-align:center;"> <tbody> <tr> <td style="text-align:inherit;padding:0;line-height:.9;font-size:inherit">n+k −1</td> </tr> <tr> <td style="text-align:inherit;padding:0;line-height:.9;font-size:inherit">k</td> </tr> </tbody> </table> ) does make sense for n = k = 0 if interpreted as a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Binomial_coefficient?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Binomial_coefficients_as_polynomials" title="Binomial coefficient">generalized binomial coefficient</a>; indeed ( <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1249493202"> <table cellspacing="0" cellpadding="0" class="RMsplit" style="text-align:center;"> <tbody> <tr> <td style="text-align:inherit;padding:0;line-height:.9;font-size:inherit">n+k −1</td> </tr> <tr> <td style="text-align:inherit;padding:0;line-height:.9;font-size:inherit">k</td> </tr> </tbody> </table> ) seen as a generalized binomial coefficient equals the extreme right-hand side of the above equation.</li> </ol> </div> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(10)"> <h2 id="References">References</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Multiset&action=edit&section=13&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: References" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-10 collapsible-block" id="mf-section-10"> <div class="mw-references-wrap mw-references-columns"> <ol class="references"> <li id="cite_note-Cantor-1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Cantor_1-0">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFCantorJourdain1895" class="citation journal cs1 cs1-prop-long-vol cs1-prop-foreign-lang-source"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Georg_Cantor?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Georg Cantor">Cantor, Georg</a>; Jourdain, Philip E.B. (Translator) (1895). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20110610133240/http://brinkmann-du.de/mathe/fos/fos01_03.htm">"beiträge zur begründung der transfiniten Mengenlehre"</a> [contributions to the founding of the theory of transfinite numbers]. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Mathematische_Annalen?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mathematische Annalen">Mathematische Annalen</a> (in German). xlvi, xlix. New York Dover Publications (1954 English translation): 481–512, 207–246. Archived from <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://brinkmann-du.de/mathe/fos/fos01_03.htm">the original</a> on 2011-06-10. <q>By a set (Menge) we are to understand any collection into a whole (Zusammenfassung zu einem Gansen) M of definite and separate objects m (p.85)</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=beitr%C3%A4ge+zur+begr%C3%BCndung+der+transfiniten+Mengenlehre&rft.volume=xlvi%3Bxlix&rft.pages=481-512%2C+207-246&rft.date=1895&rft.aulast=Cantor&rft.aufirst=Georg&rft.au=Jourdain%2C+Philip+E.B.+%28Translator%29&rft_id=http%3A%2F%2Fbrinkmann-du.de%2Fmathe%2Ffos%2Ffos01_03.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultiset" class="Z3988"></li> <li id="cite_note-2"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-2">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHein2003" class="citation book cs1">Hein, James L. (2003). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/discretemathemat00hein_966">Discrete mathematics</a>. Jones & Bartlett Publishers. pp. <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/discretemathemat00hein_966/page/n43">29</a>–30. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/0-7637-2210-3?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/0-7637-2210-3"><bdi>0-7637-2210-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Discrete+mathematics&rft.pages=29-30&rft.pub=Jones+%26+Bartlett+Publishers&rft.date=2003&rft.isbn=0-7637-2210-3&rft.aulast=Hein&rft.aufirst=James+L.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdiscretemathemat00hein_966&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultiset" class="Z3988"></li> <li id="cite_note-Knuth1998-3">^ <a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Knuth1998_3-0">a</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Knuth1998_3-1">b</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Knuth1998_3-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnuth1998" class="citation book cs1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Donald_Knuth?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Donald Knuth">Knuth, Donald E.</a> (1998). Seminumerical Algorithms. <a href="https://en-m-wikipedia-org.translate.goog/wiki/The_Art_of_Computer_Programming?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="The Art of Computer Programming">The Art of Computer Programming</a>. Vol. 2 (3rd ed.). <a href="https://en-m-wikipedia-org.translate.goog/wiki/Addison_Wesley?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Addison Wesley">Addison Wesley</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/0-201-89684-2?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/0-201-89684-2"><bdi>0-201-89684-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Seminumerical+Algorithms&rft.series=The+Art+of+Computer+Programming&rft.edition=3rd&rft.pub=Addison+Wesley&rft.date=1998&rft.isbn=0-201-89684-2&rft.aulast=Knuth&rft.aufirst=Donald+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultiset" class="Z3988"></li> <li id="cite_note-Blizard1989-4"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Blizard1989_4-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlizard1989" class="citation journal cs1">Blizard, Wayne D (1989). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://projecteuclid.org/download/pdf_1/euclid.ndjfl/1093634995">"Multiset theory"</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Notre_Dame_Journal_of_Formal_Logic?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Notre Dame Journal of Formal Logic">Notre Dame Journal of Formal Logic</a>. 30 (1): 36–66. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1305%252Fndjfl%252F1093634995">10.1305/ndjfl/1093634995</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notre+Dame+Journal+of+Formal+Logic&rft.atitle=Multiset+theory&rft.volume=30&rft.issue=1&rft.pages=36-66&rft.date=1989&rft_id=info%3Adoi%2F10.1305%2Fndjfl%2F1093634995&rft.aulast=Blizard&rft.aufirst=Wayne+D&rft_id=http%3A%2F%2Fprojecteuclid.org%2Fdownload%2Fpdf_1%2Feuclid.ndjfl%2F1093634995&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultiset" class="Z3988"></li> <li id="cite_note-Blizard1991-5">^ <a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Blizard1991_5-0">a</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Blizard1991_5-1">b</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Blizard1991_5-2">c</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Blizard1991_5-3">d</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Blizard1991_5-4">e</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlizard1991" class="citation journal cs1">Blizard, Wayne D. 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"Characteristic Functions and the Algebra of Logic". <a href="https://en-m-wikipedia-org.translate.goog/wiki/Annals_of_Mathematics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Annals of Mathematics">Annals of Mathematics</a>. 34 (3): 405–414. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.2307%252F1968168">10.2307/1968168</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/JSTOR_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.jstor.org/stable/1968168">1968168</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=Characteristic+Functions+and+the+Algebra+of+Logic&rft.volume=34&rft.issue=3&rft.pages=405-414&rft.date=1933&rft_id=info%3Adoi%2F10.2307%2F1968168&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1968168%23id-name%3DJSTOR&rft.aulast=Whitney&rft.aufirst=Hassler&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultiset" class="Z3988"></li> <li id="cite_note-15"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-15">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMonro1987" class="citation journal cs1">Monro, G. 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Zeitschrift für Mathematische Logik und Grundlagen der Mathematik. 33 (2): 171–178. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1002%252Fmalq.19870330212">10.1002/malq.19870330212</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Zeitschrift+f%C3%BCr+Mathematische+Logik+und+Grundlagen+der+Mathematik&rft.atitle=The+Concept+of+Multiset&rft.volume=33&rft.issue=2&rft.pages=171-178&rft.date=1987&rft_id=info%3Adoi%2F10.1002%2Fmalq.19870330212&rft.aulast=Monro&rft.aufirst=G.+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultiset" class="Z3988"></li> <li id="cite_note-16"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-16">^</a> Cf., for instance, Richard Brualdi, Introductory Combinatorics, Pearson.</li> <li id="cite_note-Aigner1979-18"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Aigner1979_18-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAigner1979" class="citation book cs1">Aigner, M. (1979). Combinatorial Theory. New York/Berlin: Springer Verlag.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Combinatorial+Theory&rft.place=New+York%2FBerlin&rft.pub=Springer+Verlag&rft.date=1979&rft.aulast=Aigner&rft.aufirst=M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultiset" class="Z3988"></li> <li id="cite_note-Anderson1987-19"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Anderson1987_19-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAnderson1987" class="citation book cs1">Anderson, I. (1987). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://archive.org/details/combinatoricsoff0000ande">Combinatorics of Finite Sets</a>. Oxford: Clarendon Press. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-19-853367-2?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-19-853367-2"><bdi>978-0-19-853367-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Combinatorics+of+Finite+Sets&rft.place=Oxford&rft.pub=Clarendon+Press&rft.date=1987&rft.isbn=978-0-19-853367-2&rft.aulast=Anderson&rft.aufirst=I.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcombinatoricsoff0000ande&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultiset" class="Z3988"></li> <li id="cite_note-Stanley1997-20"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Stanley1997_20-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStanley1997" class="citation book cs1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Richard_P._Stanley?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Richard P. Stanley">Stanley, Richard P.</a> (1997). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://www-math.mit.edu/~rstan/ec/">Enumerative Combinatorics</a>. Vol. 1. Cambridge University Press. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/0-521-55309-1?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/0-521-55309-1"><bdi>0-521-55309-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Enumerative+Combinatorics&rft.pub=Cambridge+University+Press&rft.date=1997&rft.isbn=0-521-55309-1&rft.aulast=Stanley&rft.aufirst=Richard+P.&rft_id=http%3A%2F%2Fwww-math.mit.edu%2F~rstan%2Fec%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultiset" class="Z3988"></li> <li id="cite_note-Stanley1999-21"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Stanley1999_21-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStanley1999" class="citation book cs1">Stanley, Richard P. (1999). Enumerative Combinatorics. Vol. 2. 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Journal of Computer and System Sciences. 52 (3): 570–588. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1006%252Fjcss.1996.0042">10.1006/jcss.1996.0042</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Computer+and+System+Sciences&rft.atitle=Towards+tractable+algebras+for+bags&rft.volume=52&rft.issue=3&rft.pages=570-588&rft.date=1996&rft_id=info%3Adoi%2F10.1006%2Fjcss.1996.0042&rft.aulast=Grumbach&rft.aufirst=S.&rft.au=Milo%2C+T&rft_id=https%3A%2F%2Fdoi.org%2F10.1006%252Fjcss.1996.0042&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultiset" class="Z3988"></li> <li id="cite_note-LibkinWong1994-23"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-LibkinWong1994_23-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLibkinWong1994" class="citation book cs1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Leonid_Libkin?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Leonid Libkin">Libkin, L.</a>; Wong, L. (1994). "Some properties of query languages for bags". Proceedings of the Workshop on Database Programming Languages. Springer Verlag. pp. 97–114.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Some+properties+of+query+languages+for+bags&rft.btitle=Proceedings+of+the+Workshop+on+Database+Programming+Languages&rft.pages=97-114&rft.pub=Springer+Verlag&rft.date=1994&rft.aulast=Libkin&rft.aufirst=L.&rft.au=Wong%2C+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultiset" class="Z3988"></li> <li id="cite_note-LibkingWong1995-24"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-LibkingWong1995_24-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLibkinWong1995" class="citation journal cs1">Libkin, L.; Wong, L. (1995). 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(1989). "Real-valued Multisets and Fuzzy Sets". <a href="https://en-m-wikipedia-org.translate.goog/wiki/Fuzzy_Sets_and_Systems?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Fuzzy Sets and Systems">Fuzzy Sets and Systems</a>. 33: 77–97. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1016%252F0165-0114%252889%252990218-2">10.1016/0165-0114(89)90218-2</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Fuzzy+Sets+and+Systems&rft.atitle=Real-valued+Multisets+and+Fuzzy+Sets&rft.volume=33&rft.pages=77-97&rft.date=1989&rft_id=info%3Adoi%2F10.1016%2F0165-0114%2889%2990218-2&rft.aulast=Blizard&rft.aufirst=Wayne+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultiset" class="Z3988"></li> <li id="cite_note-Blizard1990-26"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Blizard1990_26-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlizard1990" class="citation journal cs1">Blizard, Wayne D. (1990). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1305%252Fndjfl%252F1093635499">"Negative Membership"</a>. Notre Dame Journal of Formal Logic. 31 (1): 346–368. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1305%252Fndjfl%252F1093635499">10.1305/ndjfl/1093635499</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/S2CID_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://api.semanticscholar.org/CorpusID:42766971">42766971</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notre+Dame+Journal+of+Formal+Logic&rft.atitle=Negative+Membership&rft.volume=31&rft.issue=1&rft.pages=346-368&rft.date=1990&rft_id=info%3Adoi%2F10.1305%2Fndjfl%2F1093635499&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A42766971%23id-name%3DS2CID&rft.aulast=Blizard&rft.aufirst=Wayne+D.&rft_id=https%3A%2F%2Fdoi.org%2F10.1305%252Fndjfl%252F1093635499&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultiset" class="Z3988"></li> <li id="cite_note-Yager1986-27"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Yager1986_27-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYager1986" class="citation journal cs1">Yager, R. 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International Journal of General Systems. 13: 23–37. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1080%252F03081078608934952">10.1080/03081078608934952</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+General+Systems&rft.atitle=On+the+Theory+of+Bags&rft.volume=13&rft.pages=23-37&rft.date=1986&rft_id=info%3Adoi%2F10.1080%2F03081078608934952&rft.aulast=Yager&rft.aufirst=R.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultiset" class="Z3988"></li> <li id="cite_note-Grzymala-Busse1987-28"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Grzymala-Busse1987_28-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrzymala-Busse1987" class="citation book cs1">Grzymala-Busse, J. 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Charlotte, North Carolina. pp. 325–332.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Learning+from+examples+based+on+rough+multisets&rft.btitle=Proceedings+of+the+2nd+International+Symposium+on+Methodologies+for+Intelligent+Systems&rft.place=Charlotte%2C+North+Carolina&rft.pages=325-332&rft.date=1987&rft.aulast=Grzymala-Busse&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultiset" class="Z3988"><code class="cs1-code">{{<a href="https://en-m-wikipedia-org.translate.goog/wiki/Template:Cite_book?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: location missing publisher (<a href="https://en-m-wikipedia-org.translate.goog/wiki/Category:CS1_maint:_location_missing_publisher?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Category:CS1 maint: location missing publisher">link</a>)</li> <li id="cite_note-Loeb1992-29"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Loeb1992_29-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLoeb1992" class="citation journal cs1">Loeb, D. (1992). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1016%252F0001-8708%252892%252990011-9">"Sets with a negative numbers of elements"</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Advances_in_Mathematics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Advances in Mathematics">Advances in Mathematics</a>. 91: 64–74. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1016%252F0001-8708%252892%252990011-9">10.1016/0001-8708(92)90011-9</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Advances+in+Mathematics&rft.atitle=Sets+with+a+negative+numbers+of+elements&rft.volume=91&rft.pages=64-74&rft.date=1992&rft_id=info%3Adoi%2F10.1016%2F0001-8708%2892%2990011-9&rft.aulast=Loeb&rft.aufirst=D.&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252F0001-8708%252892%252990011-9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultiset" class="Z3988"></li> <li id="cite_note-Miyamoto2001-30"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Miyamoto2001_30-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMiyamoto2001" class="citation book cs1">Miyamoto, S. (2001). "Fuzzy Multisets and Their Generalizations". Multiset Processing. Lecture Notes in Computer Science. Vol. 2235. pp. 225–235. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1007%252F3-540-45523-X_11">10.1007/3-540-45523-X_11</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-3-540-43063-6?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-3-540-43063-6"><bdi>978-3-540-43063-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Fuzzy+Multisets+and+Their+Generalizations&rft.btitle=Multiset+Processing&rft.series=Lecture+Notes+in+Computer+Science&rft.pages=225-235&rft.date=2001&rft_id=info%3Adoi%2F10.1007%2F3-540-45523-X_11&rft.isbn=978-3-540-43063-6&rft.aulast=Miyamoto&rft.aufirst=S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultiset" class="Z3988"></li> <li id="cite_note-Alkhazaleh2011-31"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Alkhazaleh2011_31-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlkhazalehSallehHassan2011" class="citation journal cs1">Alkhazaleh, S.; Salleh, A. R.; Hassan, N. (2011). "Soft Multisets Theory". Applied Mathematical Sciences. 5 (72): 3561–3573.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Applied+Mathematical+Sciences&rft.atitle=Soft+Multisets+Theory&rft.volume=5&rft.issue=72&rft.pages=3561-3573&rft.date=2011&rft.aulast=Alkhazaleh&rft.aufirst=S.&rft.au=Salleh%2C+A.+R.&rft.au=Hassan%2C+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultiset" class="Z3988"></li> <li id="cite_note-Alkhazaleh2012-32"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-Alkhazaleh2012_32-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlkhazalehSalleh2012" class="citation journal cs1">Alkhazaleh, S.; Salleh, A. R. (2012). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1155%252F2012%252F350603">"Fuzzy Soft Multiset Theory"</a>. Abstract and Applied Analysis. 2012: 1–20. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1155%252F2012%252F350603">10.1155/2012/350603</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Abstract+and+Applied+Analysis&rft.atitle=Fuzzy+Soft+Multiset+Theory&rft.volume=2012&rft.pages=1-20&rft.date=2012&rft_id=info%3Adoi%2F10.1155%2F2012%2F350603&rft.aulast=Alkhazaleh&rft.aufirst=S.&rft.au=Salleh%2C+A.+R.&rft_id=https%3A%2F%2Fdoi.org%2F10.1155%252F2012%252F350603&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultiset" class="Z3988"></li> <li id="cite_note-33"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-33">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBurgin1990" class="citation book cs1">Burgin, Mark (1990). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://www.blogg.org/blog-30140-date-2005-10-26.html">"Theory of Named Sets as a Foundational Basis for Mathematics"</a>. Structures in Mathematical Theories. San Sebastian. pp. 417–420.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Theory+of+Named+Sets+as+a+Foundational+Basis+for+Mathematics&rft.btitle=Structures+in+Mathematical+Theories&rft.pages=417-420&rft.pub=San+Sebastian&rft.date=1990&rft.aulast=Burgin&rft.aufirst=Mark&rft_id=http%3A%2F%2Fwww.blogg.org%2Fblog-30140-date-2005-10-26.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultiset" class="Z3988"></li> <li id="cite_note-34"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-34">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBurgin1992" class="citation journal cs1">Burgin, Mark (1992). "On the concept of a multiset in cybernetics". Cybernetics and System Analysis. 3: 165–167.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Cybernetics+and+System+Analysis&rft.atitle=On+the+concept+of+a+multiset+in+cybernetics&rft.volume=3&rft.pages=165-167&rft.date=1992&rft.aulast=Burgin&rft.aufirst=Mark&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultiset" class="Z3988"></li> <li id="cite_note-35"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-35">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBurgin2004" class="citation arxiv cs1">Burgin, Mark (2004). "Unified Foundations of Mathematics". <a href="https://en-m-wikipedia-org.translate.goog/wiki/ArXiv_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://arxiv.org/abs/math/0403186">math/0403186</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Unified+Foundations+of+Mathematics&rft.date=2004&rft_id=info%3Aarxiv%2Fmath%2F0403186&rft.aulast=Burgin&rft.aufirst=Mark&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultiset" class="Z3988"></li> <li id="cite_note-36"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Multiset?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-36">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBurgin2011" class="citation book cs1">Burgin, Mark (2011). Theory of Named Sets. Mathematics Research Developments. Nova Science Pub Inc. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-1-61122-788-8?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-1-61122-788-8"><bdi>978-1-61122-788-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Named+Sets&rft.series=Mathematics+Research+Developments&rft.pub=Nova+Science+Pub+Inc&rft.date=2011&rft.isbn=978-1-61122-788-8&rft.aulast=Burgin&rft.aufirst=Mark&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMultiset" class="Z3988"></li> </ol> </div>   </section> </div> <noscript> <img src="https://login.m.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&mobile=1" alt="" width="1" height="1" style="border: none; position: absolute;"> </noscript> <div 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mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ca.wikipedia.org/wiki/Multiconjunt" title="Multiconjunt – Catalan" lang="ca" hreflang="ca" data-title="Multiconjunt" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li> <li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://cs.wikipedia.org/wiki/Multimno%25C5%25BEina" title="Multimnožina – Czech" lang="cs" hreflang="cs" data-title="Multimnožina" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li> <li class="interlanguage-link interwiki-de mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://de.wikipedia.org/wiki/Multimenge" title="Multimenge – German" lang="de" hreflang="de" data-title="Multimenge" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li> <li class="interlanguage-link interwiki-es mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://es.wikipedia.org/wiki/Multiconjunto" title="Multiconjunto – Spanish" lang="es" hreflang="es" data-title="Multiconjunto" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li> <li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://eo.wikipedia.org/wiki/Multaro" title="Multaro – Esperanto" lang="eo" hreflang="eo" data-title="Multaro" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li> <li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://eu.wikipedia.org/wiki/Multimultzo" title="Multimultzo – Basque" lang="eu" hreflang="eu" data-title="Multimultzo" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li> <li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fa.wikipedia.org/wiki/%25DA%2586%25D9%2586%25D8%25AF%25D9%2585%25D8%25AC%25D9%2585%25D9%2588%25D8%25B9%25D9%2587" title="چندمجموعه – Persian" lang="fa" hreflang="fa" data-title="چندمجموعه" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li> <li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fr.wikipedia.org/wiki/Multiensemble" title="Multiensemble – French" lang="fr" hreflang="fr" data-title="Multiensemble" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li> <li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ko.wikipedia.org/wiki/%25EC%25A4%2591%25EB%25B3%25B5%25EC%25A7%2591%25ED%2595%25A9" title="중복집합 – Korean" lang="ko" hreflang="ko" data-title="중복집합" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li> <li class="interlanguage-link interwiki-it mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://it.wikipedia.org/wiki/Multiinsieme" title="Multiinsieme – Italian" lang="it" hreflang="it" data-title="Multiinsieme" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li> <li class="interlanguage-link interwiki-he mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://he.wikipedia.org/wiki/%25D7%259E%25D7%2595%25D7%259C%25D7%2598%25D7%2599_%25D7%25A7%25D7%2591%25D7%2595%25D7%25A6%25D7%2594" title="מולטי קבוצה – Hebrew" lang="he" hreflang="he" data-title="מולטי קבוצה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li> <li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://nl.wikipedia.org/wiki/Multiset" title="Multiset – Dutch" lang="nl" hreflang="nl" data-title="Multiset" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li> <li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ja.wikipedia.org/wiki/%25E5%25A4%259A%25E9%2587%258D%25E9%259B%2586%25E5%2590%2588" title="多重集合 – Japanese" lang="ja" hreflang="ja" data-title="多重集合" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li> <li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pl.wikipedia.org/wiki/Multizbi%25C3%25B3r" title="Multizbiór – Polish" lang="pl" hreflang="pl" data-title="Multizbiór" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li> <li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://pt.wikipedia.org/wiki/Multiconjunto" title="Multiconjunto – Portuguese" lang="pt" hreflang="pt" data-title="Multiconjunto" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li> <li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ro.wikipedia.org/wiki/Multimul%25C8%259Bime" title="Multimulțime – Romanian" lang="ro" hreflang="ro" data-title="Multimulțime" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li> <li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://ru.wikipedia.org/wiki/%25D0%259C%25D1%2583%25D0%25BB%25D1%258C%25D1%2582%25D0%25B8%25D0%25BC%25D0%25BD%25D0%25BE%25D0%25B6%25D0%25B5%25D1%2581%25D1%2582%25D0%25B2%25D0%25BE" title="Мультимножество – Russian" lang="ru" hreflang="ru" data-title="Мультимножество" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li> <li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://simple.wikipedia.org/wiki/Multiset" title="Multiset – Simple English" lang="en-simple" hreflang="en-simple" data-title="Multiset" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li> <li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sl.wikipedia.org/wiki/Ve%25C4%258Dkratna_mno%25C5%25BEica" title="Večkratna množica – Slovenian" lang="sl" hreflang="sl" data-title="Večkratna množica" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li> <li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sv.wikipedia.org/wiki/Multim%25C3%25A4ngd" title="Multimängd – Swedish" lang="sv" hreflang="sv" 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