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id="siteNotice"></div> </div> <div class="pre-content heading-holder"> <div class="page-heading"> <h1 id="firstHeading" class="firstHeading mw-first-heading">Set (mathematics)</h1> <div class="tagline"></div> </div> <ul id="p-associated-pages" class="minerva__tab-container"> <li class="minerva__tab selected"> <a class="minerva__tab-text" href="/wiki/Set_(mathematics)" rel="" data-event-name="tabs.subject">Article</a> </li> <li class="minerva__tab "> <a class="minerva__tab-text" href="/wiki/Talk:Set_(mathematics)" rel="discussion" data-event-name="tabs.talk">Talk</a> </li> </ul> <nav class="page-actions-menu"> <ul id="p-views" class="page-actions-menu__list"> <li id="language-selector" class="page-actions-menu__list-item"> <a role="button" href="#p-lang" data-mw="interface" data-event-name="menu.languages" title="Language" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet language-selector"> Language </a> </li> <li id="page-actions-watch" class="page-actions-menu__list-item"> <a role="button" id="ca-watch" href="/w/index.php?title=Special:UserLogin&returnto=Set+%28mathematics%29" data-event-name="menu.watch" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet menu__item--page-actions-watch"> Watch </a> </li> <li id="page-actions-edit" class="page-actions-menu__list-item"> <a role="button" id="ca-edit" href="/w/index.php?title=Set_(mathematics)&action=edit" data-event-name="menu.edit" data-mw="interface" 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-page menu__item--page-actions-edit"> Edit </a> </li> </ul> </nav>  <div id="mw-content-subtitle"></div> </div> <div id="bodyContent" class="content"> <div id="mw-content-text" class="mw-body-content"><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 what mathematicians call "intuitive" or "naive" set theory. For a more detailed account, see <a href="/wiki/Naive_set_theory" title="Naive set theory">Naive set theory</a>. For a rigorous modern axiomatic treatment of sets, see <a href="/wiki/Set_theory" title="Set theory">Set theory</a>.</div> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a set is a collection of different<a href="#cite_note-Cantor-1">[1]</a> things;<a href="#cite_note-JainAhmad1995-2">[2]</a><a href="#cite_note-Goldberg1986-3">[3]</a><a href="#cite_note-CormenCormen2001-4">[4]</a> these things are called <a href="/wiki/Element_(mathematics)" title="Element (mathematics)">elements</a> or members of the set and are typically <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical objects</a> of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets.<a href="#cite_note-FOOTNOTEHalmos1960%5Bhttpsarchiveorgdetailsnaivesettheory00halmpagen11mode2up_1%5D-5">[5]</a> A set may have a finite number of elements or be an <a href="/wiki/Infinite_set" title="Infinite set">infinite set</a>. There is a unique set with no elements, called the <a href="/wiki/Empty_set" title="Empty set">empty set</a>; a set with a single element is a <a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">singleton</a>. <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Example_of_a_set.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Example_of_a_set.svg/220px-Example_of_a_set.svg.png" decoding="async" width="220" height="196" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Example_of_a_set.svg/330px-Example_of_a_set.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/Example_of_a_set.svg/440px-Example_of_a_set.svg.png 2x" data-file-width="450" data-file-height="400"></a><figcaption>A set of polygons in an <a href="/wiki/Euler_diagram" title="Euler diagram">Euler diagram</a></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Example_of_a_set_rearranged.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Example_of_a_set_rearranged.svg/220px-Example_of_a_set_rearranged.svg.png" decoding="async" width="220" height="196" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Example_of_a_set_rearranged.svg/330px-Example_of_a_set_rearranged.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/05/Example_of_a_set_rearranged.svg/440px-Example_of_a_set_rearranged.svg.png 2x" data-file-width="450" data-file-height="400"></a><figcaption>This set equals the one depicted above since both have the very same elements.</figcaption></figure> Sets are uniquely characterized by their elements; this means that two sets that have precisely the same elements are <a href="/wiki/Equality_(mathematics)" title="Equality (mathematics)">equal</a> (they are the same set).<a href="#cite_note-Stoll-6">[6]</a> This property is called <a href="/wiki/Axiom_of_extensionality" title="Axiom of extensionality">extensionality</a>. In particular, this implies that there is only one empty set. Sets are ubiquitous in modern mathematics. Indeed, <a href="/wiki/Set_theory" title="Set theory">set theory</a>, more specifically <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel set theory</a>, has been the standard way to provide rigorous <a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">foundations</a> for all branches of mathematics since the first half of the 20th century.<a href="#cite_note-FOOTNOTEHalmos1960%5Bhttpsarchiveorgdetailsnaivesettheory00halmpagen11mode2up_1%5D-5">[5]</a> <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="#Definition_and_notation">1 Definition and notation</a> <ul> <li class="toclevel-2 tocsection-2"><a href="#Roster_notation">1.1 Roster notation</a> <ul> <li class="toclevel-3 tocsection-3"><a href="#Infinite_sets_in_roster_notation">1.1.1 Infinite sets in roster notation</a></li> </ul> </li> <li class="toclevel-2 tocsection-4"><a href="#Semantic_definition">1.2 Semantic definition</a></li> <li class="toclevel-2 tocsection-5"><a href="#Set-builder_notation">1.3 Set-builder notation</a></li> <li class="toclevel-2 tocsection-6"><a href="#Classifying_methods_of_definition">1.4 Classifying methods of definition</a></li> </ul> </li> <li class="toclevel-1 tocsection-7"><a href="#Membership">2 Membership</a></li> <li class="toclevel-1 tocsection-8"><a href="#The_empty_set">3 The empty set</a></li> <li class="toclevel-1 tocsection-9"><a href="#Singleton_sets">4 Singleton sets</a></li> <li class="toclevel-1 tocsection-10"><a href="#Subsets">5 Subsets</a></li> <li class="toclevel-1 tocsection-11"><a href="#Euler_and_Venn_diagrams">6 Euler and Venn diagrams</a></li> <li class="toclevel-1 tocsection-12"><a href="#Special_sets_of_numbers_in_mathematics">7 Special sets of numbers in mathematics</a></li> <li class="toclevel-1 tocsection-13"><a href="#Functions">8 Functions</a></li> <li class="toclevel-1 tocsection-14"><a href="#Cardinality">9 Cardinality</a> <ul> <li class="toclevel-2 tocsection-15"><a href="#Infinite_sets_and_infinite_cardinality">9.1 Infinite sets and infinite cardinality</a></li> <li class="toclevel-2 tocsection-16"><a href="#The_continuum_hypothesis">9.2 The continuum hypothesis</a></li> </ul> </li> <li class="toclevel-1 tocsection-17"><a href="#Power_sets">10 Power sets</a></li> <li class="toclevel-1 tocsection-18"><a href="#Partitions">11 Partitions</a></li> <li class="toclevel-1 tocsection-19"><a href="#Basic_operations">12 Basic operations</a></li> <li class="toclevel-1 tocsection-20"><a href="#Applications">13 Applications</a></li> <li class="toclevel-1 tocsection-21"><a href="#Principle_of_inclusion_and_exclusion">14 Principle of inclusion and exclusion</a></li> <li class="toclevel-1 tocsection-22"><a href="#History">15 History</a> <ul> <li class="toclevel-2 tocsection-23"><a href="#Naive_set_theory">15.1 Naive set theory</a></li> <li class="toclevel-2 tocsection-24"><a href="#Axiomatic_set_theory">15.2 Axiomatic set theory</a></li> </ul> </li> <li class="toclevel-1 tocsection-25"><a href="#See_also">16 See also</a></li> <li class="toclevel-1 tocsection-26"><a href="#Notes">17 Notes</a></li> <li class="toclevel-1 tocsection-27"><a href="#References">18 References</a></li> <li class="toclevel-1 tocsection-28"><a href="#External_links">19 External links</a></li> </ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"><h2 id="Definition_and_notation">Definition and notation </h2> <a role="button" href="/w/index.php?title=Set_(mathematics)&action=edit&section=1" title="Edit section: Definition and notation" 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"> Mathematical texts commonly denote sets by <a href="/wiki/Capital_letters" class="mw-redirect" title="Capital letters">capital letters</a><a href="#cite_note-LipschutzLipson1997-7">[7]</a><a href="#cite_note-FOOTNOTEHalmos1960%5Bhttpsarchiveorgdetailsnaivesettheory00halmpagen11mode2up_1%5D-5">[5]</a> in <a href="/wiki/Italic_type" title="Italic type">italic</a>, such as A, B, C.<a href="#cite_note-:1-8">[8]</a> A set may also be called a collection or family, especially when its elements are themselves sets. <div class="mw-heading mw-heading3"><h3 id="Roster_notation">Roster notation</h3> <a role="button" href="/w/index.php?title=Set_(mathematics)&action=edit&section=2" title="Edit section: Roster notation" 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> Roster or enumeration notation defines a set by listing its elements between <a href="/wiki/Curly_bracket" class="mw-redirect" title="Curly bracket">curly brackets</a>, separated by commas:<a href="#cite_note-Roberts2009-9">[9]</a><a href="#cite_note-JohnsonJohnson2004-10">[10]</a><a href="#cite_note-BelloKaul2013-11">[11]</a><a href="#cite_note-Epp2010-12">[12]</a> <style data-mw-deduplicate="TemplateStyles:r996643573">.mw-parser-output .block-indent{padding-left:3em;padding-right:0;overflow:hidden}</style><div class="block-indent">A = {4, 2, 1, 3}</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573"><div class="block-indent">B = {blue, white, red}.</div> This notation was introduced by <a href="/wiki/Ernst_Zermelo" title="Ernst Zermelo">Ernst Zermelo</a> in 1908.<a href="#cite_note-13">[13]</a> In a set, all that matters is whether each element is in it or not, so the ordering of the elements in roster notation is irrelevant (in contrast, in a <a href="/wiki/Sequence" title="Sequence">sequence</a>, a <a href="/wiki/Tuple" title="Tuple">tuple</a>, or a <a href="/wiki/Permutation" title="Permutation">permutation</a> of a set, the ordering of the terms matters). For example, {2, 4, 6} and {4, 6, 4, 2} represent the same set.<a href="#cite_note-MaurerRalston2005-14">[14]</a><a href="#cite_note-:1-8">[8]</a><a href="#cite_note-DalenDoets2014-15">[15]</a> For sets with many elements, especially those following an implicit pattern, the list of members can be abbreviated using an <a href="/wiki/Ellipsis#In_mathematical_notation" title="Ellipsis">ellipsis</a> '...'.<a href="#cite_note-BastaDeLong2013-16">[16]</a><a href="#cite_note-BrackenMiller2013-17">[17]</a> For instance, the set of the first thousand positive integers may be specified in roster notation as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573"><div class="block-indent">{1, 2, 3, ..., 1000}.</div> <div class="mw-heading mw-heading4"><h4 id="Infinite_sets_in_roster_notation">Infinite sets in roster notation</h4> <a role="button" href="/w/index.php?title=Set_(mathematics)&action=edit&section=3" title="Edit section: Infinite sets in roster notation" 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> An <a href="/wiki/Infinite_set" title="Infinite set">infinite set</a> is a set with an infinite number of elements. If the pattern of its elements is obvious, an infinite set can be given in roster notation, with an ellipsis placed at the end of the list, or at both ends, to indicate that the list continues forever. For example, the set of <a href="/wiki/Nonnegative_integer" class="mw-redirect" title="Nonnegative integer">nonnegative integers</a> is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573"><div class="block-indent">{0, 1, 2, 3, 4, ...},</div> and the set of all <a href="/wiki/Integer" title="Integer">integers</a> is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573"><div class="block-indent">{..., −3, −2, −1, 0, 1, 2, 3, ...}.</div> <div class="mw-heading mw-heading3"><h3 id="Semantic_definition">Semantic definition</h3> <a role="button" href="/w/index.php?title=Set_(mathematics)&action=edit&section=4" title="Edit section: Semantic 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> Another way to define a set is to use a rule to determine what the elements are: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573"><div class="block-indent">Let A be the set whose members are the first four positive <a href="/wiki/Integer" title="Integer">integers</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573"><div class="block-indent">Let B be the set of colors of the <a href="/wiki/Flag_of_France" title="Flag of France">French flag</a>.</div> Such a definition is called a semantic description.<a href="#cite_note-FOOTNOTEHalmos1960%5Bhttpsarchiveorgdetailsnaivesettheory00halmpage4mode2up_4%5D-18">[18]</a><a href="#cite_note-Ruda2011-19">[19]</a> <div class="mw-heading mw-heading3"><h3 id="Set-builder_notation">Set-builder notation</h3> <a role="button" href="/w/index.php?title=Set_(mathematics)&action=edit&section=5" title="Edit section: Set-builder notation" 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> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Set-builder_notation" title="Set-builder notation">Set-builder notation</a></div> Set-builder notation specifies a set as a selection from a larger set, determined by a condition on the elements.<a href="#cite_note-Ruda2011-19">[19]</a><a href="#cite_note-Lucas1990-20">[20]</a><a href="#cite_note-21">[21]</a> For example, a set F can be defined as follows: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=\{n\mid n{\text{ is an integer, and }}0\leq n\leq 19\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>n</mi> <mo>∣</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is an integer, and </mtext> </mrow> <mn>0</mn> <mo>≤</mo> <mi>n</mi> <mo>≤</mo> <mn>19</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=\{n\mid n{\text{ is an integer, and }}0\leq n\leq 19\}.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/477b3e7e7d57f892acaf78f63ced9e376cca5f55" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.915ex; height:2.843ex;" alt="{\displaystyle F=\{n\mid n{\text{ is an integer, and }}0\leq n\leq 19\}.}"></noscript><span class="lazy-image-placeholder" style="width: 41.915ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/477b3e7e7d57f892acaf78f63ced9e376cca5f55" data-alt="{\displaystyle F=\{n\mid n{\text{ is an integer, and }}0\leq n\leq 19\}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  In this notation, the <a href="/wiki/Vertical_bar" title="Vertical bar">vertical bar</a> "|" means "such that", and the description can be interpreted as "F is the set of all numbers n such that n is an integer in the range from 0 to 19 inclusive". Some authors use a <a href="/wiki/Colon_(punctuation)" title="Colon (punctuation)">colon</a> ":" instead of the vertical bar.<a href="#cite_note-Steinlage1987-22">[22]</a> <div class="mw-heading mw-heading3"><h3 id="Classifying_methods_of_definition">Classifying methods of definition</h3> <a role="button" href="/w/index.php?title=Set_(mathematics)&action=edit&section=6" title="Edit section: Classifying methods of 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> <a href="/wiki/Philosophy" title="Philosophy">Philosophy</a> uses specific terms to classify types of definitions: <ul><li>An <a href="/wiki/Intensional_definition" class="mw-redirect" title="Intensional definition">intensional definition</a> uses a rule to determine membership. Semantic definitions and definitions using set-builder notation are examples.</li> <li>An <a href="/wiki/Extensional_definition" class="mw-redirect" title="Extensional definition">extensional definition</a> describes a set by listing all its elements.<a href="#cite_note-Ruda2011-19">[19]</a> Such definitions are also called <a href="/wiki/Enumerative_definition" title="Enumerative definition">enumerative</a>.</li> <li>An <a href="/wiki/Ostensive_definition" title="Ostensive definition">ostensive definition</a> is one that describes a set by giving examples of elements; a roster involving an ellipsis would be an example.</li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"><h2 id="Membership">Membership</h2> <a role="button" href="/w/index.php?title=Set_(mathematics)&action=edit&section=7" title="Edit section: Membership" 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"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Element (mathematics)</a></div> If B is a set and x is an element of B, this is written in shorthand as x ∈ B, which can also be read as "x belongs to B", or "x is in B".<a href="#cite_note-FOOTNOTEHalmos1960%5Bhttpsarchiveorgdetailsnaivesettheory00halmpage2mode2up_2%5D-23">[23]</a> The statement "y is not an element of B" is written as y ∉ B, which can also be read as "y is not in B".<a href="#cite_note-CapinskiKopp2004-24">[24]</a><a href="#cite_note-25">[25]</a> For example, with respect to the sets A = {1, 2, 3, 4}, B = {blue, white, red}, and F = {n | n is an integer, and 0 ≤ n ≤ 19}, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573"><div class="block-indent">4 ∈ A and 12 ∈ F; and</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573"><div class="block-indent">20 ∉ F and green ∉ B.</div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"><h2 id="The_empty_set">The empty set</h2> <a role="button" href="/w/index.php?title=Set_(mathematics)&action=edit&section=8" title="Edit section: The empty set" 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"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Empty_set" title="Empty set">Empty set</a></div> The empty set (or null set) is the unique set that has no members. It is denoted ∅, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \emptyset }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \emptyset }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6af50205f42bb2ec3c666b7b847d2c7f96e464c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \emptyset }"></noscript><span class="lazy-image-placeholder" style="width: 1.162ex;height: 2.509ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6af50205f42bb2ec3c666b7b847d2c7f96e464c7" data-alt="{\displaystyle \emptyset }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , { },<a href="#cite_note-FOOTNOTEHalmos1960%5Bhttpsarchiveorgdetailsnaivesettheory00halmpage8mode2up_8%5D-26">[26]</a><a href="#cite_note-LeungChen1992-27">[27]</a> ϕ,<a href="#cite_note-28">[28]</a> or ϕ.<a href="#cite_note-29">[29]</a> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"><h2 id="Singleton_sets">Singleton sets</h2> <a role="button" href="/w/index.php?title=Set_(mathematics)&action=edit&section=9" title="Edit section: Singleton sets" 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"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton (mathematics)</a></div> A singleton set is a set with exactly one element; such a set may also be called a unit set.<a href="#cite_note-Stoll-6">[6]</a> Any such set can be written as {x}, where x is the element. The set {x} and the element x mean different things; Halmos<a href="#cite_note-FOOTNOTEHalmos1960%5Bhttpsarchiveorgdetailsnaivesettheory00halmpage4mode2up_Sect.2%5D-30">[30]</a> draws the analogy that a box containing a hat is not the same as the hat. </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"><h2 id="Subsets">Subsets</h2> <a role="button" href="/w/index.php?title=Set_(mathematics)&action=edit&section=10" title="Edit section: Subsets" 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"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Subset" title="Subset">Subset</a></div> If every element of set A is also in B, then A is described as being a subset of B, or contained in B, written A ⊆ B,<a href="#cite_note-Hausdorff2005-31">[31]</a> or B ⊇ A.<a href="#cite_note-Comninos2010-32">[32]</a> The latter notation may be read B contains A, B includes A, or B is a superset of A. The <a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">relationship</a> between sets established by ⊆ is called inclusion or containment. Two sets are equal if they contain each other: A ⊆ B and B ⊆ A is equivalent to A = B.<a href="#cite_note-Lucas1990-20">[20]</a> If A is a subset of B, but A is not equal to B, then A is called a proper subset of B. This can be written A ⊊ B. Likewise, B ⊋ A means B is a proper superset of A, i.e. B contains A, and is not equal to A. A third pair of operators ⊂ and ⊃ are used differently by different authors: some authors use A ⊂ B and B ⊃ A to mean A is any subset of B (and not necessarily a proper subset),<a href="#cite_note-FOOTNOTEHalmos1960%5Bhttpsarchiveorgdetailsnaivesettheory00halmpage2mode2up_3%5D-33">[33]</a><a href="#cite_note-CapinskiKopp2004-24">[24]</a> while others reserve A ⊂ B and B ⊃ A for cases where A is a proper subset of B.<a href="#cite_note-Hausdorff2005-31">[31]</a> Examples: <ul><li>The set of all humans is a proper subset of the set of all mammals.</li> <li>{1, 3} ⊂ {1, 2, 3, 4}.</li> <li>{1, 2, 3, 4} ⊆ {1, 2, 3, 4}.</li></ul> The empty set is a subset of every set,<a href="#cite_note-FOOTNOTEHalmos1960%5Bhttpsarchiveorgdetailsnaivesettheory00halmpage8mode2up_8%5D-26">[26]</a> and every set is a subset of itself:<a href="#cite_note-FOOTNOTEHalmos1960%5Bhttpsarchiveorgdetailsnaivesettheory00halmpage2mode2up_3%5D-33">[33]</a> <ul><li>∅ ⊆ A.</li> <li>A ⊆ A.</li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"><h2 id="Euler_and_Venn_diagrams">Euler and Venn diagrams</h2> <a role="button" href="/w/index.php?title=Set_(mathematics)&action=edit&section=11" title="Edit section: Euler and Venn diagrams" 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"> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Venn_A_subset_B.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Venn_A_subset_B.svg/150px-Venn_A_subset_B.svg.png" decoding="async" width="150" height="150" class="mw-file-element" data-file-width="155" data-file-height="155"></noscript><span class="lazy-image-placeholder" style="width: 150px;height: 150px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Venn_A_subset_B.svg/150px-Venn_A_subset_B.svg.png" data-width="150" data-height="150" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Venn_A_subset_B.svg/225px-Venn_A_subset_B.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b0/Venn_A_subset_B.svg/300px-Venn_A_subset_B.svg.png 2x" data-class="mw-file-element"> </a><figcaption>A is a subset of B. B is a superset of A.</figcaption></figure> An <a href="/wiki/Euler_diagram" title="Euler diagram">Euler diagram</a> is a graphical representation of a collection of sets; each set is depicted as a planar region enclosed by a loop, with its elements inside. If A is a subset of B, then the region representing A is completely inside the region representing B. If two sets have no elements in common, the regions do not overlap. A <a href="/wiki/Venn_diagram" title="Venn diagram">Venn diagram</a>, in contrast, is a graphical representation of n sets in which the n loops divide the plane into 2n zones such that for each way of selecting some of the n sets (possibly all or none), there is a zone for the elements that belong to all the selected sets and none of the others. For example, if the sets are A, B, and C, there should be a zone for the elements that are inside A and C and outside B (even if such elements do not exist). </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"><h2 id="Special_sets_of_numbers_in_mathematics">Special sets of numbers in mathematics</h2> <a role="button" href="/w/index.php?title=Set_(mathematics)&action=edit&section=12" title="Edit section: Special sets of numbers in mathematics" 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"> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:NumberSetinC.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/NumberSetinC.svg/220px-NumberSetinC.svg.png" decoding="async" width="220" height="172" class="mw-file-element" data-file-width="600" data-file-height="470"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 172px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/NumberSetinC.svg/220px-NumberSetinC.svg.png" data-width="220" data-height="172" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/NumberSetinC.svg/330px-NumberSetinC.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a0/NumberSetinC.svg/440px-NumberSetinC.svg.png 2x" data-class="mw-file-element"> </a><figcaption>The <a href="/wiki/Natural_numbers" class="mw-redirect" title="Natural numbers">natural numbers</a> <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">  are contained in the <a href="/wiki/Integers" class="mw-redirect" title="Integers">integers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" data-alt="{\displaystyle \mathbb {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , which are contained in the <a href="/wiki/Rational_numbers" class="mw-redirect" title="Rational numbers">rational numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></noscript><span class="lazy-image-placeholder" style="width: 1.808ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" data-alt="{\displaystyle \mathbb {Q} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , which are contained in the <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" 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 {R} }"></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/786849c765da7a84dbc3cce43e96aad58a5868dc" data-alt="{\displaystyle \mathbb {R} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , which are contained in the <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" 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 {C} }"></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/f9add4085095b9b6d28d045fd9c92c2c09f549a7" data-alt="{\displaystyle \mathbb {C} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </figcaption></figure> There are sets of such mathematical importance, to which mathematicians refer so frequently, that they have acquired special names and notational conventions to identify them. Many of these important sets are represented in mathematical texts using bold (e.g. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b776aaf12c2da4b78ca777cb8295c2000bfd51f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.634ex; height:2.176ex;" alt="{\displaystyle \mathbf {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 1.634ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b776aaf12c2da4b78ca777cb8295c2000bfd51f5" data-alt="{\displaystyle \mathbf {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ) or <a href="/wiki/Blackboard_bold" title="Blackboard bold">blackboard bold</a> (e.g. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" data-alt="{\displaystyle \mathbb {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ) typeface.<a href="#cite_note-Tourlakis2003-34">[34]</a> These include <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {N} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2f63b6cd6d63ee9b7be0b7e4d14099d7153bd43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.091ex; height:2.176ex;" alt="{\displaystyle \mathbf {N} }"></noscript><span class="lazy-image-placeholder" style="width: 2.091ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2f63b6cd6d63ee9b7be0b7e4d14099d7153bd43" data-alt="{\displaystyle \mathbf {N} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  or <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"> , the set of all <a href="/wiki/Natural_number" title="Natural number">natural numbers</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {N} =\{0,1,2,3,...\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {N} =\{0,1,2,3,...\}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4e56e443c36d1169793ec83704be5b091096fba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.402ex; height:2.843ex;" alt="{\displaystyle \mathbf {N} =\{0,1,2,3,...\}}"></noscript><span class="lazy-image-placeholder" style="width: 19.402ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4e56e443c36d1169793ec83704be5b091096fba" data-alt="{\displaystyle \mathbf {N} =\{0,1,2,3,...\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  (often, authors exclude 0);<a href="#cite_note-Tourlakis2003-34">[34]</a></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b776aaf12c2da4b78ca777cb8295c2000bfd51f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.634ex; height:2.176ex;" alt="{\displaystyle \mathbf {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 1.634ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b776aaf12c2da4b78ca777cb8295c2000bfd51f5" data-alt="{\displaystyle \mathbf {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></noscript><span class="lazy-image-placeholder" style="width: 1.55ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" data-alt="{\displaystyle \mathbb {Z} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , the set of all <a href="/wiki/Integer" title="Integer">integers</a> (whether positive, negative or zero): <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Z} =\{...,-2,-1,0,1,2,3,...\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Z} =\{...,-2,-1,0,1,2,3,...\}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc04b5e7bcf1d9a5b99026491b203989de97b6f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.09ex; height:2.843ex;" alt="{\displaystyle \mathbf {Z} =\{...,-2,-1,0,1,2,3,...\}}"></noscript><span class="lazy-image-placeholder" style="width: 31.09ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc04b5e7bcf1d9a5b99026491b203989de97b6f0" data-alt="{\displaystyle \mathbf {Z} =\{...,-2,-1,0,1,2,3,...\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> ;<a href="#cite_note-Tourlakis2003-34">[34]</a></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132d0144479d6f47c30ad82a65d458966ccbe928" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.008ex; height:2.509ex;" alt="{\displaystyle \mathbf {Q} }"></noscript><span class="lazy-image-placeholder" style="width: 2.008ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132d0144479d6f47c30ad82a65d458966ccbe928" data-alt="{\displaystyle \mathbf {Q} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></noscript><span class="lazy-image-placeholder" style="width: 1.808ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" data-alt="{\displaystyle \mathbb {Q} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , the set of all <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> (that is, the set of all <a href="/wiki/Proper_fraction" class="mw-redirect" title="Proper fraction">proper</a> and <a href="/wiki/Improper_fraction" class="mw-redirect" title="Improper fraction">improper fractions</a>): <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} =\left\{{\frac {a}{b}}\mid a,b\in \mathbf {Z} ,b\neq 0\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>∣</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo>,</mo> <mi>b</mi> <mo>≠</mo> <mn>0</mn> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} =\left\{{\frac {a}{b}}\mid a,b\in \mathbf {Z} ,b\neq 0\right\}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5dd8d6f8bfd25dc94c7f5819bc780c74715d211" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.238ex; height:5.009ex;" alt="{\displaystyle \mathbf {Q} =\left\{{\frac {a}{b}}\mid a,b\in \mathbf {Z} ,b\neq 0\right\}}"></noscript><span class="lazy-image-placeholder" style="width: 26.238ex;height: 5.009ex;vertical-align: -2.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5dd8d6f8bfd25dc94c7f5819bc780c74715d211" data-alt="{\displaystyle \mathbf {Q} =\left\{{\frac {a}{b}}\mid a,b\in \mathbf {Z} ,b\neq 0\right\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . For example, −<style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠7/4⁠ ∈ Q and 5 = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035">⁠5/1⁠ ∈ Q;<a href="#cite_note-Tourlakis2003-34">[34]</a></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }"></noscript><span class="lazy-image-placeholder" style="width: 2.003ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" data-alt="{\displaystyle \mathbf {R} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" 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 {R} }"></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/786849c765da7a84dbc3cce43e96aad58a5868dc" data-alt="{\displaystyle \mathbb {R} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , the set of all <a href="/wiki/Real_number" title="Real number">real numbers</a>, including all rational numbers and all <a href="/wiki/Irrational_number" title="Irrational number">irrational</a> numbers (which include <a href="/wiki/Algebraic_number" title="Algebraic number">algebraic numbers</a> such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></noscript><span class="lazy-image-placeholder" style="width: 3.098ex;height: 3.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" data-alt="{\displaystyle {\sqrt {2}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  that cannot be rewritten as fractions, as well as <a href="/wiki/Transcendental_numbers" class="mw-redirect" title="Transcendental numbers">transcendental numbers</a> such as <a href="/wiki/Pi" title="Pi">π</a> and <a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e</a>);<a href="#cite_note-Tourlakis2003-34">[34]</a></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {C} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle \mathbf {C} }"></noscript><span class="lazy-image-placeholder" style="width: 1.931ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11de80478fce9090e43eed19100b37cc841661e8" data-alt="{\displaystyle \mathbf {C} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" 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 {C} }"></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/f9add4085095b9b6d28d045fd9c92c2c09f549a7" data-alt="{\displaystyle \mathbb {C} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , the set of all <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>: C = {a + bi | a, b ∈ R}, for example, 1 + 2i ∈ C.<a href="#cite_note-Tourlakis2003-34">[34]</a></li></ul> Each of the above sets of numbers has an infinite number of elements. Each is a subset of the sets listed below it. Sets of positive or negative numbers are sometimes denoted by superscript plus and minus signs, respectively. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} ^{+}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9fafdbdb9efd367bcb05e6e310a9c85326281bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.519ex; height:2.843ex;" alt="{\displaystyle \mathbf {Q} ^{+}}"></noscript><span class="lazy-image-placeholder" style="width: 3.519ex;height: 2.843ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9fafdbdb9efd367bcb05e6e310a9c85326281bb" data-alt="{\displaystyle \mathbf {Q} ^{+}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  represents the set of positive rational numbers. </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(8)"><h2 id="Functions">Functions</h2> <a role="button" href="/w/index.php?title=Set_(mathematics)&action=edit&section=13" title="Edit section: Functions" 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"> A <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> (or <a href="/wiki/Map_(mathematics)" title="Map (mathematics)">mapping</a>) from a set A to a set B is a rule that assigns to each "input" element of A an "output" that is an element of B; more formally, a function is a special kind of <a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">relation</a>, one that relates each element of A to exactly one element of B. A function is called <ul><li><a href="/wiki/Injective" class="mw-redirect" title="Injective">injective</a> (or one-to-one) if it maps any two different elements of A to different elements of B,</li> <li><a href="/wiki/Surjective" class="mw-redirect" title="Surjective">surjective</a> (or onto) if for every element of B, there is at least one element of A that maps to it, and</li> <li><a href="/wiki/Bijective" class="mw-redirect" title="Bijective">bijective</a> (or a one-to-one correspondence) if the function is both injective and surjective — in this case, each element of A is paired with a unique element of B, and each element of B is paired with a unique element of A, so that there are no unpaired elements.</li></ul> An injective function is called an injection, a surjective function is called a surjection, and a bijective function is called a bijection or one-to-one correspondence. </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(9)"><h2 id="Cardinality">Cardinality</h2> <a role="button" href="/w/index.php?title=Set_(mathematics)&action=edit&section=14" title="Edit section: Cardinality" 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"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Cardinality" title="Cardinality">Cardinality</a></div> The cardinality of a set S, denoted |S|, is the number of members of S.<a href="#cite_note-Moschovakis1994-35">[35]</a> For example, if B = {blue, white, red}, then |B| = 3. Repeated members in roster notation are not counted,<a href="#cite_note-Fleck2001-36">[36]</a><a href="#cite_note-Johnston2015-37">[37]</a> so |{blue, white, red, blue, white}| = 3, too. More formally, two sets share the same cardinality if there exists a bijection between them. The cardinality of the empty set is zero.<a href="#cite_note-Smith2008-38">[38]</a> <div class="mw-heading mw-heading3"><h3 id="Infinite_sets_and_infinite_cardinality">Infinite sets and infinite cardinality</h3> <a role="button" href="/w/index.php?title=Set_(mathematics)&action=edit&section=15" title="Edit section: Infinite sets and infinite cardinality" 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 list of elements of some sets is endless, or <a href="/wiki/Infinite_set" title="Infinite set">infinite</a>. For example, 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 <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> is infinite.<a href="#cite_note-Lucas1990-20">[20]</a> In fact, all the special sets of numbers mentioned in the section above are infinite. Infinite sets have infinite cardinality. Some infinite cardinalities are greater than others. Arguably one of the most significant results from set theory is that the set of <a href="/wiki/Real_number" title="Real number">real numbers</a> has greater cardinality than the set of natural numbers.<a href="#cite_note-Stillwell2013-39">[39]</a> Sets with cardinality less than or equal to that of <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">  are called <a href="/wiki/Countable_set" title="Countable set">countable sets</a>; these are either finite sets or <a href="/wiki/Countably_infinite_set" class="mw-redirect" title="Countably infinite set">countably infinite sets</a> (sets of the same cardinality as <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"> ); some authors use "countable" to mean "countably infinite". Sets with cardinality strictly greater than that of <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">  are called <a href="/wiki/Uncountable_set" title="Uncountable set">uncountable sets</a>. However, it can be shown that the cardinality of a <a href="/wiki/Straight_line" class="mw-redirect" title="Straight line">straight line</a> (i.e., the number of points on a line) is the same as the cardinality of any <a href="/wiki/Line_segment" title="Line segment">segment</a> of that line, of the entire <a href="/wiki/Plane_(mathematics)" title="Plane (mathematics)">plane</a>, and indeed of any <a href="/wiki/Dimension_(mathematics)" class="mw-redirect" title="Dimension (mathematics)">finite-dimensional</a> <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>.<a href="#cite_note-Tall2006-40">[40]</a> <div class="mw-heading mw-heading3"><h3 id="The_continuum_hypothesis">The continuum hypothesis</h3> <a role="button" href="/w/index.php?title=Set_(mathematics)&action=edit&section=16" title="Edit section: The continuum hypothesis" 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> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">Continuum hypothesis</a></div> The continuum hypothesis, formulated by Georg Cantor in 1878, is the statement that there is no set with cardinality strictly between the <a href="/wiki/Aleph-nought" class="mw-redirect" title="Aleph-nought">cardinality of the natural numbers</a> and the cardinality of a straight line.<a href="#cite_note-Cantor1878-41">[41]</a> In 1963, <a href="/wiki/Paul_Cohen" title="Paul Cohen">Paul Cohen</a> proved that the continuum hypothesis is <a href="/wiki/Independence_(mathematical_logic)" title="Independence (mathematical logic)">independent</a> of the axiom system ZFC consisting of <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel set theory</a> with the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a>.<a href="#cite_note-Cohen1963-42">[42]</a> (ZFC is the most widely-studied version of axiomatic set theory.) </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(10)"><h2 id="Power_sets">Power sets</h2> <a role="button" href="/w/index.php?title=Set_(mathematics)&action=edit&section=17" title="Edit section: Power sets" 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"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Power_set" title="Power set">Power set</a></div> The power set of a set S is the set of all subsets of S.<a href="#cite_note-Lucas1990-20">[20]</a> The <a href="/wiki/Empty_set" title="Empty set">empty set</a> and S itself are elements of the power set of S, because these are both subsets of S. For example, the power set of {1, 2, 3} is {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}. The power set of a set S is commonly written as P(S) or 2S.<a href="#cite_note-Lucas1990-20">[20]</a><a href="#cite_note-FOOTNOTEHalmos1960%5Bhttpsarchiveorgdetailsnaivesettheory00halmpage18mode2up_19%5D-43">[43]</a><a href="#cite_note-:1-8">[8]</a> If S has n elements, then P(S) has 2n elements.<a href="#cite_note-FOOTNOTEHalmos1960%5Bhttpsarchiveorgdetailsnaivesettheory00halmpage20mode2up_20%5D-44">[44]</a> For example, {1, 2, 3} has three elements, and its power set has 23 = 8 elements, as shown above. If S is infinite (whether <a href="/wiki/Countable" class="mw-redirect" title="Countable">countable</a> or <a href="/wiki/Uncountable" class="mw-redirect" title="Uncountable">uncountable</a>), then P(S) is uncountable. Moreover, the power set is always strictly "bigger" than the original set, in the sense that any attempt to pair up the elements of S with the elements of P(S) will leave some elements of P(S) unpaired. (There is never a <a href="/wiki/Bijection" title="Bijection">bijection</a> from S onto P(S).)<a href="#cite_note-BurgerStarbird2004-45">[45]</a> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(11)"><h2 id="Partitions">Partitions</h2> <a role="button" href="/w/index.php?title=Set_(mathematics)&action=edit&section=18" title="Edit section: Partitions" 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-11 collapsible-block" id="mf-section-11"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Partition_of_a_set" title="Partition of a set">Partition of a set</a></div> A <a href="/wiki/Partition_of_a_set" title="Partition of a set">partition of a set</a> S is a set of nonempty subsets of S, such that every element x in S is in exactly one of these subsets. That is, the subsets are <a href="/wiki/Pairwise_disjoint" class="mw-redirect" title="Pairwise disjoint">pairwise disjoint</a> (meaning any two sets of the partition contain no element in common), and the <a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a> of all the subsets of the partition is S.<a href="#cite_note-Mansour2012-46">[46]</a><a href="#cite_note-FOOTNOTEHalmos1960%5Bhttpsarchiveorgdetailsnaivesettheory00halmpage28mode2up_28%5D-47">[47]</a> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(12)"><h2 id="Basic_operations">Basic operations</h2> <a role="button" href="/w/index.php?title=Set_(mathematics)&action=edit&section=19" title="Edit section: Basic 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-12 collapsible-block" id="mf-section-12"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Algebra_of_sets" title="Algebra of sets">Algebra of sets</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Venn1010.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Venn1010.svg/220px-Venn1010.svg.png" decoding="async" width="220" height="162" class="mw-file-element" data-file-width="380" data-file-height="280"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 162px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Venn1010.svg/220px-Venn1010.svg.png" data-width="220" data-height="162" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Venn1010.svg/330px-Venn1010.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Venn1010.svg/440px-Venn1010.svg.png 2x" data-class="mw-file-element"> </a><figcaption><div class="center">The complement of A in U</div></figcaption></figure> Suppose that a <a href="/wiki/Universe_(mathematics)" title="Universe (mathematics)">universal set</a> U (a set containing all elements being discussed) has been fixed, and that A is a subset of U. <ul><li>The <a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complement</a> of A is the set of all elements (of U) that do not belong to A. It may be denoted Ac or A′. In set-builder notation, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{\text{c}}=\{a\in U:a\notin A\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>c</mtext> </mrow> </msup> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo>∈</mo> <mi>U</mi> <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^{\text{c}}=\{a\in U:a\notin A\}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0701aea496ff8e874e2498f71b03935b469c9071" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.733ex; height:2.843ex;" alt="{\displaystyle A^{\text{c}}=\{a\in U:a\notin A\}}"></noscript><span class="lazy-image-placeholder" style="width: 21.733ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0701aea496ff8e874e2498f71b03935b469c9071" data-alt="{\displaystyle A^{\text{c}}=\{a\in U:a\notin A\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . The complement may also be called the absolute complement to distinguish it from the relative complement below. Example: If the universal set is taken to be the set of integers, then the complement of the set of even integers is the set of odd integers.</li></ul> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Venn0111.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Venn0111.svg/220px-Venn0111.svg.png" decoding="async" width="220" height="162" class="mw-file-element" data-file-width="380" data-file-height="280"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 162px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Venn0111.svg/220px-Venn0111.svg.png" data-width="220" data-height="162" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Venn0111.svg/330px-Venn0111.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/30/Venn0111.svg/440px-Venn0111.svg.png 2x" data-class="mw-file-element"> </a><figcaption><div class="center">The union of A and B, denoted A ∪ B</div></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Venn0001.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/220px-Venn0001.svg.png" decoding="async" width="220" height="160" class="mw-file-element" data-file-width="410" data-file-height="299"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 160px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/220px-Venn0001.svg.png" data-width="220" data-height="160" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/330px-Venn0001.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Venn0001.svg/440px-Venn0001.svg.png 2x" data-class="mw-file-element"> </a><figcaption><div class="center">The intersection of A and B, denoted A ∩ B</div></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Venn0100.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Venn0100.svg/220px-Venn0100.svg.png" decoding="async" width="220" height="162" class="mw-file-element" data-file-width="380" data-file-height="280"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 162px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Venn0100.svg/220px-Venn0100.svg.png" data-width="220" data-height="162" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Venn0100.svg/330px-Venn0100.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Venn0100.svg/440px-Venn0100.svg.png 2x" data-class="mw-file-element"> </a><figcaption><div class="center">The set difference A \ B</div></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Venn0110.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Venn0110.svg/220px-Venn0110.svg.png" decoding="async" width="220" height="160" class="mw-file-element" data-file-width="410" data-file-height="299"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 160px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Venn0110.svg/220px-Venn0110.svg.png" data-width="220" data-height="160" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/46/Venn0110.svg/330px-Venn0110.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/46/Venn0110.svg/440px-Venn0110.svg.png 2x" data-class="mw-file-element"> </a><figcaption><div class="center">The symmetric difference of A and B</div></figcaption></figure> Given any two sets A and B, <ul><li>their <a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a> A ∪ B is the set of all things that are members of A or B or both.</li> <li>their <a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a> A ∩ B is the set of all things that are members of both A and B. If A ∩ B = ∅, then A and B are said to be disjoint.</li> <li>the <a href="/wiki/Set_difference" class="mw-redirect" title="Set difference">set difference</a> A \ B (also written A − B) is the set of all things that belong to A but not B. Especially when B is a subset of A, it is also called the <a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">relative complement</a> of B in A. With Bc as the absolute complement of B (in the universal set U), A \ B = A ∩ Bc .</li> <li>their <a href="/wiki/Symmetric_difference" title="Symmetric difference">symmetric difference</a> A Δ B is the set of all things that belong to A or B but not both. One has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\,\Delta \,B=(A\setminus B)\cup (B\setminus A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal">Δ</mi> <mspace width="thinmathspace"></mspace> <mi>B</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo class="MJX-variant">∖</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>∪</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo class="MJX-variant">∖</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\,\Delta \,B=(A\setminus B)\cup (B\setminus A)}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fde641f23775b76a25efd5aa3536c4f520770d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.92ex; height:2.843ex;" alt="{\displaystyle A\,\Delta \,B=(A\setminus B)\cup (B\setminus A)}"></noscript><span class="lazy-image-placeholder" style="width: 26.92ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fde641f23775b76a25efd5aa3536c4f520770d3f" data-alt="{\displaystyle A\,\Delta \,B=(A\setminus B)\cup (B\setminus A)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> .</li> <li>their <a href="/wiki/Cartesian_product" title="Cartesian product">cartesian product</a> A × B is the set of all <a href="/wiki/Ordered_pairs" class="mw-redirect" title="Ordered pairs">ordered pairs</a> (a,b) such that a is an element of A and b is an element of B.</li></ul> Examples: <ul><li>{1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}.</li> <li>{1, 2, 3} ∩ {3, 4, 5} = {3}.</li> <li>{1, 2, 3} − {3, 4, 5} = {1, 2}.</li> <li>{1, 2, 3} Δ {3, 4, 5} = {1, 2, 4, 5}.</li> <li>{a, b} × {1, 2, 3} = {(a,1), (a,2), (a,3), (b,1), (b,2), (b,3)}.</li></ul> The operations above satisfy many identities. For example, one of <a href="/wiki/De_Morgan%27s_laws" title="De Morgan's laws">De Morgan's laws</a> states that (A ∪ B)′ = A′ ∩ B′ (that is, the elements outside the union of A and B are the elements that are outside A and outside B). The cardinality of A × B is the product of the cardinalities of A and B. This is an elementary fact when A and B are finite. When one or both are infinite, multiplication of cardinal numbers is defined to make this true. The power set of any set becomes a <a href="/wiki/Boolean_ring" title="Boolean ring">Boolean ring</a> with symmetric difference as the addition of the ring and intersection as the multiplication of the ring. </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(13)"><h2 id="Applications">Applications</h2> <a role="button" href="/w/index.php?title=Set_(mathematics)&action=edit&section=20" 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-13 collapsible-block" id="mf-section-13"> Sets are ubiquitous in modern mathematics. For example, <a href="/wiki/Algebraic_structure" title="Algebraic structure">structures</a> in <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a>, such as <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a>, <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a> and <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">rings</a>, are sets <a href="/wiki/Closure_(mathematics)" title="Closure (mathematics)">closed</a> under one or more operations. One of the main applications of naive set theory is in the construction of <a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">relations</a>. A relation from a <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> A to a <a href="/wiki/Codomain" title="Codomain">codomain</a> B is a subset of the Cartesian product A × B. For example, considering the set S = {rock, paper, scissors} of shapes in the <a href="/wiki/Rock_paper_scissors" title="Rock paper scissors">game</a> of the same name, the relation "beats" from S to S is the set B = {(scissors,paper), (paper,rock), (rock,scissors)}; thus x beats y in the game if the pair (x,y) is a member of B. Another example is the set F of all pairs (x, x2), where x is real. This relation is a subset of R × R, because the set of all squares is subset of the set of all real numbers. Since for every x in R, one and only one pair (x,...) is found in F, it is called a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a>. In functional notation, this relation can be written as F(x) = x2. </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(14)"><h2 id="Principle_of_inclusion_and_exclusion">Principle of inclusion and exclusion</h2> <a role="button" href="/w/index.php?title=Set_(mathematics)&action=edit&section=21" title="Edit section: Principle of inclusion and exclusion" 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-14 collapsible-block" id="mf-section-14"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Inclusion%E2%80%93exclusion_principle" title="Inclusion–exclusion principle">Inclusion–exclusion principle</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:A_union_B.svg" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/A_union_B.svg/220px-A_union_B.svg.png" decoding="async" width="220" height="164" class="mw-file-element" data-file-width="226" data-file-height="168"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 164px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/A_union_B.svg/220px-A_union_B.svg.png" data-width="220" data-height="164" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/A_union_B.svg/330px-A_union_B.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/86/A_union_B.svg/440px-A_union_B.svg.png 2x" data-class="mw-file-element"> </a><figcaption>The inclusion-exclusion principle for two finite sets states that the size of their union is the sum of the sizes of the sets minus the size of their intersection.</figcaption></figure> The inclusion–exclusion principle is a technique for counting the elements in a union of two finite sets in terms of the sizes of the two sets and their intersection. It can be expressed symbolically as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |A\cup B|=|A|+|B|-|A\cap B|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mo>∪</mo> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <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> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mo>∩</mo> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |A\cup B|=|A|+|B|-|A\cap B|.}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0bbdac1eb429befc47b8dca8180078642af9592" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.287ex; height:2.843ex;" alt="{\displaystyle |A\cup B|=|A|+|B|-|A\cap B|.}"></noscript><span class="lazy-image-placeholder" style="width: 30.287ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0bbdac1eb429befc47b8dca8180078642af9592" data-alt="{\displaystyle |A\cup B|=|A|+|B|-|A\cap B|.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  A more general form of the principle gives the cardinality of any finite union of finite sets: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\left|A_{1}\cup A_{2}\cup A_{3}\cup \ldots \cup A_{n}\right|=&\left(\left|A_{1}\right|+\left|A_{2}\right|+\left|A_{3}\right|+\ldots \left|A_{n}\right|\right)\\&{}-\left(\left|A_{1}\cap A_{2}\right|+\left|A_{1}\cap A_{3}\right|+\ldots \left|A_{n-1}\cap A_{n}\right|\right)\\&{}+\ldots \\&{}+\left(-1\right)^{n-1}\left(\left|A_{1}\cap A_{2}\cap A_{3}\cap \ldots \cap A_{n}\right|\right).\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="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow> <mo>|</mo> <mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∪</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∪</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>∪</mo> <mo>…</mo> <mo>∪</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>|</mo> </mrow> <mo>=</mo> </mtd> <mtd> <mrow> <mo>(</mo> <mrow> <mrow> <mo>|</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>|</mo> </mrow> <mo>+</mo> <mrow> <mo>|</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>|</mo> </mrow> <mo>+</mo> <mrow> <mo>|</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>|</mo> </mrow> <mo>+</mo> <mo>…</mo> <mrow> <mo>|</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>|</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mrow> <mo>|</mo> <mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∩</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>|</mo> </mrow> <mo>+</mo> <mrow> <mo>|</mo> <mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∩</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mo>|</mo> </mrow> <mo>+</mo> <mo>…</mo> <mrow> <mo>|</mo> <mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>∩</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>|</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>+</mo> <mo>…</mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mo>|</mo> <mrow> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∩</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∩</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>∩</mo> <mo>…</mo> <mo>∩</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>|</mo> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left|A_{1}\cup A_{2}\cup A_{3}\cup \ldots \cup A_{n}\right|=&\left(\left|A_{1}\right|+\left|A_{2}\right|+\left|A_{3}\right|+\ldots \left|A_{n}\right|\right)\\&{}-\left(\left|A_{1}\cap A_{2}\right|+\left|A_{1}\cap A_{3}\right|+\ldots \left|A_{n-1}\cap A_{n}\right|\right)\\&{}+\ldots \\&{}+\left(-1\right)^{n-1}\left(\left|A_{1}\cap A_{2}\cap A_{3}\cap \ldots \cap A_{n}\right|\right).\end{aligned}}}</annotation> </semantics> </math><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/784bb31a02981a4f1854676abd7cdf7094bfa196" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:73.188ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}\left|A_{1}\cup A_{2}\cup A_{3}\cup \ldots \cup A_{n}\right|=&\left(\left|A_{1}\right|+\left|A_{2}\right|+\left|A_{3}\right|+\ldots \left|A_{n}\right|\right)\\&{}-\left(\left|A_{1}\cap A_{2}\right|+\left|A_{1}\cap A_{3}\right|+\ldots \left|A_{n-1}\cap A_{n}\right|\right)\\&{}+\ldots \\&{}+\left(-1\right)^{n-1}\left(\left|A_{1}\cap A_{2}\cap A_{3}\cap \ldots \cap A_{n}\right|\right).\end{aligned}}}"></noscript><span class="lazy-image-placeholder" style="width: 73.188ex;height: 12.509ex;vertical-align: -5.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/784bb31a02981a4f1854676abd7cdf7094bfa196" data-alt="{\displaystyle {\begin{aligned}\left|A_{1}\cup A_{2}\cup A_{3}\cup \ldots \cup A_{n}\right|=&\left(\left|A_{1}\right|+\left|A_{2}\right|+\left|A_{3}\right|+\ldots \left|A_{n}\right|\right)\\&{}-\left(\left|A_{1}\cap A_{2}\right|+\left|A_{1}\cap A_{3}\right|+\ldots \left|A_{n-1}\cap A_{n}\right|\right)\\&{}+\ldots \\&{}+\left(-1\right)^{n-1}\left(\left|A_{1}\cap A_{2}\cap A_{3}\cap \ldots \cap A_{n}\right|\right).\end{aligned}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(15)"><h2 id="History">History</h2> <a role="button" href="/w/index.php?title=Set_(mathematics)&action=edit&section=22" 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-15 collapsible-block" id="mf-section-15"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Set_theory" title="Set theory">Set theory</a></div> The concept of a set emerged in mathematics at the end of the 19th century.<a href="#cite_note-Ferreir%C3%B3s2007-48">[48]</a> The German word for set, Menge, was coined by <a href="/wiki/Bernard_Bolzano" title="Bernard Bolzano">Bernard Bolzano</a> in his work <a href="/wiki/Paradoxes_of_the_Infinite" title="Paradoxes of the Infinite">Paradoxes of the Infinite</a>.<a href="#cite_note-Russ2004-49">[49]</a><a href="#cite_note-EwaldEwald1996-50">[50]</a><a href="#cite_note-RusnockSebest%C3%ADk2019-51">[51]</a><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Passage_with_the_set_definition_of_Georg_Cantor.png" class="mw-file-description"><noscript><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Passage_with_the_set_definition_of_Georg_Cantor.png/220px-Passage_with_the_set_definition_of_Georg_Cantor.png" decoding="async" width="220" height="93" class="mw-file-element" data-file-width="1401" data-file-height="594"></noscript><span class="lazy-image-placeholder" style="width: 220px;height: 93px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Passage_with_the_set_definition_of_Georg_Cantor.png/220px-Passage_with_the_set_definition_of_Georg_Cantor.png" data-width="220" data-height="93" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Passage_with_the_set_definition_of_Georg_Cantor.png/330px-Passage_with_the_set_definition_of_Georg_Cantor.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/Passage_with_the_set_definition_of_Georg_Cantor.png/440px-Passage_with_the_set_definition_of_Georg_Cantor.png 2x" data-class="mw-file-element"> </a><figcaption>Passage with a translation of the original set definition of Georg Cantor. The German word Menge for set is translated with aggregate here.</figcaption></figure> <a href="/wiki/Georg_Cantor" title="Georg Cantor">Georg Cantor</a>, one of the founders of set theory, gave the following definition at the beginning of his Beiträge zur Begründung der transfiniten Mengenlehre:<a href="#cite_note-Cantor.1895-52">[52]</a><a href="#cite_note-Cantor-1">[1]</a> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote">A set is a gathering together into a whole of definite, distinct objects of our perception or our thought—which are called elements of the set.</blockquote> <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a> introduced the distinction between a set and a <a href="/wiki/Class_(mathematics)" class="mw-redirect" title="Class (mathematics)">class</a> (a set is a class, but some classes, such as the class of all sets, are not sets; see <a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a>):<a href="#cite_note-53">[53]</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote">When mathematicians deal with what they call a manifold, aggregate, Menge, ensemble, or some equivalent name, it is common, especially where the number of terms involved is finite, to regard the object in question (which is in fact a class) as defined by the enumeration of its terms, and as consisting possibly of a single term, which in that case is the class.</blockquote> <div class="mw-heading mw-heading3"><h3 id="Naive_set_theory">Naive set theory</h3> <a role="button" href="/w/index.php?title=Set_(mathematics)&action=edit&section=23" title="Edit section: Naive set theory" 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> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Naive_set_theory" title="Naive set theory">Naive set theory</a></div> The foremost property of a set is that it can have elements, also called members. Two sets are <a href="/wiki/Equality_(mathematics)" title="Equality (mathematics)">equal</a> when they have the same elements. More precisely, sets A and B are equal if every element of A is an element of B, and every element of B is an element of A; this property is called the <a href="/wiki/Extensionality#In_mathematics" title="Extensionality">extensionality</a> of sets.<a href="#cite_note-FOOTNOTEHalmos1960%5Bhttpsarchiveorgdetailsnaivesettheory00halmpage2mode2up_2%5D-23">[23]</a> As a consequence, e.g. {2, 4, 6} and {4, 6, 4, 2} represent the same set. Unlike sets, <a href="/wiki/Multiset" title="Multiset">multisets</a> can be distinguished by the number of occurrences of an element; e.g. [2, 4, 6] and [4, 6, 4, 2] represent different multisets, while [2, 4, 6] and [6, 4, 2] are equal. <a href="/wiki/Tuple" title="Tuple">Tuples</a> can even be distinguished by element order; e.g. (2, 4, 6) and (6, 4, 2) represent different tuples. The simple concept of a set has proved enormously useful in mathematics, but <a href="/wiki/Category:Paradoxes_of_naive_set_theory" title="Category:Paradoxes of naive set theory">paradoxes</a> arise if no restrictions are placed on how sets can be constructed: <ul><li><a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's paradox</a> shows that the "set of all sets that do not contain themselves", i.e., {x | x is a set and x ∉ x}, cannot exist.</li> <li><a href="/wiki/Cantor%27s_paradox" title="Cantor's paradox">Cantor's paradox</a> shows that "the set of all sets" cannot exist.</li></ul> <a href="/wiki/Naive_set_theory" title="Naive set theory">Naïve set theory</a> defines a set as any <a href="/wiki/Well-defined" class="mw-redirect" title="Well-defined">well-defined</a> collection of distinct elements, but problems arise from the vagueness of the term well-defined. <div class="mw-heading mw-heading3"><h3 id="Axiomatic_set_theory">Axiomatic set theory</h3> <a role="button" href="/w/index.php?title=Set_(mathematics)&action=edit&section=24" title="Edit section: Axiomatic set theory" 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> In subsequent efforts to resolve these paradoxes since the time of the original formulation of naïve set theory, the properties of sets have been defined by <a href="/wiki/Axioms" class="mw-redirect" title="Axioms">axioms</a>. <a href="/wiki/Axiomatic_set_theory" class="mw-redirect" title="Axiomatic set theory">Axiomatic set theory</a> takes the concept of a set as a <a href="/wiki/Primitive_notion" title="Primitive notion">primitive notion</a>.<a href="#cite_note-Ferreiros2001-54">[54]</a> The purpose of the axioms is to provide a basic framework from which to deduce the truth or falsity of particular <a href="/wiki/Proposition_(mathematics)" class="mw-redirect" title="Proposition (mathematics)">mathematical propositions</a> (statements) about sets, using <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a>. According to <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">Gödel's incompleteness theorems</a> however, it is not possible to use first-order logic to prove any such particular axiomatic set theory is free from paradox.<a href="#cite_note-55">[55]</a> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(16)"><h2 id="See_also">See also</h2> <a role="button" href="/w/index.php?title=Set_(mathematics)&action=edit&section=25" 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-16 collapsible-block" id="mf-section-16"> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 20em;"> <ul><li><a href="/wiki/Algebra_of_sets" title="Algebra of sets">Algebra of sets</a></li> <li><a href="/wiki/Alternative_set_theory" class="mw-redirect" title="Alternative set theory">Alternative set theory</a></li> <li><a href="/wiki/Category_of_sets" title="Category of sets">Category of sets</a></li> <li><a href="/wiki/Class_(set_theory)" title="Class (set theory)">Class (set theory)</a></li> <li><a href="/wiki/Family_of_sets" title="Family of sets">Family of sets</a></li> <li><a href="/wiki/Fuzzy_set" title="Fuzzy set">Fuzzy set</a></li> <li><a href="/wiki/Mereology" title="Mereology">Mereology</a></li> <li><a href="/wiki/Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></li></ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(17)"><h2 id="Notes">Notes</h2> <a role="button" href="/w/index.php?title=Set_(mathematics)&action=edit&section=26" 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-17 collapsible-block" id="mf-section-17"> <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"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Cantor-1">^ <a href="#cite_ref-Cantor_1-0">a</a> <a href="#cite_ref-Cantor_1-1">b</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="CITEREFCantorJourdain1915" class="citation book cs1">Cantor, Georg; Jourdain, Philip E.B. (Translator) (1915). Contributions to the founding of the theory of transfinite numbers. New York Dover Publications (1954 English translation). <q>By an 'aggregate' (Menge) we are to understand any collection into a whole (Zusammenfassung zu einem Ganzen) M of definite and separate objects m of our intuition or our thought.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Contributions+to+the+founding+of+the+theory+of+transfinite+numbers&rft.pub=New+York+Dover+Publications+%281954+English+translation%29&rft.date=1915&rft.aulast=Cantor&rft.aufirst=Georg&rft.au=Jourdain%2C+Philip+E.B.+%28Translator%29&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> Here: p.85 </li> <li id="cite_note-JainAhmad1995-2"><a href="#cite_ref-JainAhmad1995_2-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFP._K._JainKhalil_AhmadOm_P._Ahuja1995" class="citation book cs1">P. K. Jain; Khalil Ahmad; Om P. Ahuja (1995). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yZ68h97pnAkC&pg=PA1">Functional Analysis</a>. New Age International. p. 1. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-81-224-0801-0" title="Special:BookSources/978-81-224-0801-0"><bdi>978-81-224-0801-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Functional+Analysis&rft.pages=1&rft.pub=New+Age+International&rft.date=1995&rft.isbn=978-81-224-0801-0&rft.au=P.+K.+Jain&rft.au=Khalil+Ahmad&rft.au=Om+P.+Ahuja&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DyZ68h97pnAkC%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-Goldberg1986-3"><a href="#cite_ref-Goldberg1986_3-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSamuel_Goldberg1986" class="citation book cs1">Samuel Goldberg (1 January 1986). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CmzFx9rB_FcC&pg=PA2">Probability: An Introduction</a>. Courier Corporation. p. 2. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-65252-8" title="Special:BookSources/978-0-486-65252-8"><bdi>978-0-486-65252-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability%3A+An+Introduction&rft.pages=2&rft.pub=Courier+Corporation&rft.date=1986-01-01&rft.isbn=978-0-486-65252-8&rft.au=Samuel+Goldberg&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DCmzFx9rB_FcC%26pg%3DPA2&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-CormenCormen2001-4"><a href="#cite_ref-CormenCormen2001_4-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThomas_H._CormenCharles_E_LeisersonRonald_L_RivestClifford_Stein2001" class="citation book cs1">Thomas H. Cormen; Charles E Leiserson; Ronald L Rivest; Clifford Stein (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=NLngYyWFl_YC&pg=PA1070">Introduction To Algorithms</a>. MIT Press. p. 1070. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-03293-3" title="Special:BookSources/978-0-262-03293-3"><bdi>978-0-262-03293-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+To+Algorithms&rft.pages=1070&rft.pub=MIT+Press&rft.date=2001&rft.isbn=978-0-262-03293-3&rft.au=Thomas+H.+Cormen&rft.au=Charles+E+Leiserson&rft.au=Ronald+L+Rivest&rft.au=Clifford+Stein&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DNLngYyWFl_YC%26pg%3DPA1070&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-FOOTNOTEHalmos1960[httpsarchiveorgdetailsnaivesettheory00halmpagen11mode2up_1]-5">^ <a href="#cite_ref-FOOTNOTEHalmos1960%5Bhttpsarchiveorgdetailsnaivesettheory00halmpagen11mode2up_1%5D_5-0">a</a> <a href="#cite_ref-FOOTNOTEHalmos1960%5Bhttpsarchiveorgdetailsnaivesettheory00halmpagen11mode2up_1%5D_5-1">b</a> <a href="#cite_ref-FOOTNOTEHalmos1960%5Bhttpsarchiveorgdetailsnaivesettheory00halmpagen11mode2up_1%5D_5-2">c</a> <a href="#CITEREFHalmos1960">Halmos 1960</a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/naivesettheory00halm/page/n11/mode/2up">1</a>. </li> <li id="cite_note-Stoll-6">^ <a href="#cite_ref-Stoll_6-0">a</a> <a href="#cite_ref-Stoll_6-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStoll1974" class="citation book cs1">Stoll, Robert (1974). <a rel="nofollow" class="external text" href="https://archive.org/details/setslogicaxiomat0000stol">Sets, Logic and Axiomatic Theories</a>. W. H. Freeman and Company. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/setslogicaxiomat0000stol/page/5">5</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780716704577" title="Special:BookSources/9780716704577"><bdi>9780716704577</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sets%2C+Logic+and+Axiomatic+Theories&rft.pages=5&rft.pub=W.+H.+Freeman+and+Company&rft.date=1974&rft.isbn=9780716704577&rft.aulast=Stoll&rft.aufirst=Robert&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsetslogicaxiomat0000stol&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-LipschutzLipson1997-7"><a href="#cite_ref-LipschutzLipson1997_7-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSeymor_LipschutzMarc_Lipson1997" class="citation book cs1">Seymor Lipschutz; Marc Lipson (22 June 1997). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6A5g3RiYiBUC&pg=PA1">Schaum's Outline of Discrete Mathematics</a>. McGraw Hill Professional. p. 1. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-136841-4" title="Special:BookSources/978-0-07-136841-4"><bdi>978-0-07-136841-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Schaum%27s+Outline+of+Discrete+Mathematics&rft.pages=1&rft.pub=McGraw+Hill+Professional&rft.date=1997-06-22&rft.isbn=978-0-07-136841-4&rft.au=Seymor+Lipschutz&rft.au=Marc+Lipson&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D6A5g3RiYiBUC%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-:1-8">^ <a href="#cite_ref-:1_8-0">a</a> <a href="#cite_ref-:1_8-1">b</a> <a href="#cite_ref-:1_8-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathsisfun.com/sets/sets-introduction.html">"Introduction to Sets"</a>. www.mathsisfun.com. 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CRC Press. p. 45. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4200-6956-3" title="Special:BookSources/978-1-4200-6956-3"><bdi>978-1-4200-6956-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Mathematical+Proofs%3A+A+Transition&rft.pages=45&rft.pub=CRC+Press&rft.date=2009-06-24&rft.isbn=978-1-4200-6956-3&rft.au=Charles+Roberts&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DNjBLnLyE4jAC%26pg%3DPA45&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-JohnsonJohnson2004-10"><a href="#cite_ref-JohnsonJohnson2004_10-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavid_JohnsonDavid_B._JohnsonThomas_A._Mowry2004" class="citation book cs1">David Johnson; David B. Johnson; Thomas A. Mowry (June 2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZQAqzxLFXhoC&pg=PA220">Finite Mathematics: Practical Applications (Docutech Version)</a>. W. H. 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Britton (29 January 2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=d8Se_8DWTQ4C&pg=PA47">Topics in Contemporary Mathematics</a>. Cengage Learning. p. 47. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-133-10742-2" title="Special:BookSources/978-1-133-10742-2"><bdi>978-1-133-10742-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topics+in+Contemporary+Mathematics&rft.pages=47&rft.pub=Cengage+Learning&rft.date=2013-01-29&rft.isbn=978-1-133-10742-2&rft.au=Ignacio+Bello&rft.au=Anton+Kaul&rft.au=Jack+R.+Britton&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dd8Se_8DWTQ4C%26pg%3DPA47&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-Epp2010-12"><a href="#cite_ref-Epp2010_12-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSusanna_S._Epp2010" class="citation book cs1">Susanna S. 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CRC Press. p. 11. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4398-6375-6" title="Special:BookSources/978-1-4398-6375-6"><bdi>978-1-4398-6375-6</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+Algorithmic+Mathematics&rft.pages=11&rft.pub=CRC+Press&rft.date=2005-01-21&rft.isbn=978-1-4398-6375-6&rft.au=Stephen+B.+Maurer&rft.au=Anthony+Ralston&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D_0vNBQAAQBAJ%26pg%3DPA11&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-DalenDoets2014-15"><a href="#cite_ref-DalenDoets2014_15-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFD._Van_DalenH._C._DoetsH._De_Swart2014" class="citation book cs1">D. Van Dalen; H. C. Doets; H. De Swart (9 May 2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PfbiBQAAQBAJ&pg=PA1">Sets: Naïve, Axiomatic and Applied: A Basic Compendium with Exercises for Use in Set Theory for Non Logicians, Working and Teaching Mathematicians and Students</a>. Elsevier Science. p. 1. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4831-5039-0" title="Special:BookSources/978-1-4831-5039-0"><bdi>978-1-4831-5039-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sets%3A+Na%C3%AFve%2C+Axiomatic+and+Applied%3A+A+Basic+Compendium+with+Exercises+for+Use+in+Set+Theory+for+Non+Logicians%2C+Working+and+Teaching+Mathematicians+and+Students&rft.pages=1&rft.pub=Elsevier+Science&rft.date=2014-05-09&rft.isbn=978-1-4831-5039-0&rft.au=D.+Van+Dalen&rft.au=H.+C.+Doets&rft.au=H.+De+Swart&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DPfbiBQAAQBAJ%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-BastaDeLong2013-16"><a href="#cite_ref-BastaDeLong2013_16-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlfred_BastaStephan_DeLongNadine_Basta2013" class="citation book cs1">Alfred Basta; Stephan DeLong; Nadine Basta (1 January 2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VUYLAAAAQBAJ&pg=PA3">Mathematics for Information Technology</a>. Cengage Learning. p. 3. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-285-60843-3" title="Special:BookSources/978-1-285-60843-3"><bdi>978-1-285-60843-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+for+Information+Technology&rft.pages=3&rft.pub=Cengage+Learning&rft.date=2013-01-01&rft.isbn=978-1-285-60843-3&rft.au=Alfred+Basta&rft.au=Stephan+DeLong&rft.au=Nadine+Basta&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DVUYLAAAAQBAJ%26pg%3DPA3&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-BrackenMiller2013-17"><a href="#cite_ref-BrackenMiller2013_17-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLaura_BrackenEd_Miller2013" class="citation book cs1">Laura Bracken; Ed Miller (15 February 2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nFkrl_kDiTAC&pg=PA36">Elementary Algebra</a>. Cengage Learning. p. 36. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-618-95134-5" title="Special:BookSources/978-0-618-95134-5"><bdi>978-0-618-95134-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Algebra&rft.pages=36&rft.pub=Cengage+Learning&rft.date=2013-02-15&rft.isbn=978-0-618-95134-5&rft.au=Laura+Bracken&rft.au=Ed+Miller&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DnFkrl_kDiTAC%26pg%3DPA36&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-FOOTNOTEHalmos1960[httpsarchiveorgdetailsnaivesettheory00halmpage4mode2up_4]-18"><a href="#cite_ref-FOOTNOTEHalmos1960%5Bhttpsarchiveorgdetailsnaivesettheory00halmpage4mode2up_4%5D_18-0">^</a> <a href="#CITEREFHalmos1960">Halmos 1960</a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/naivesettheory00halm/page/4/mode/2up">4</a>. </li> <li id="cite_note-Ruda2011-19">^ <a href="#cite_ref-Ruda2011_19-0">a</a> <a href="#cite_ref-Ruda2011_19-1">b</a> <a href="#cite_ref-Ruda2011_19-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrank_Ruda2011" class="citation book cs1">Frank Ruda (6 October 2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VV0SBwAAQBAJ&pg=PA151">Hegel's Rabble: An Investigation into Hegel's Philosophy of Right</a>. 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Hong Kong University Press. p. 27. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-962-209-026-2" title="Special:BookSources/978-962-209-026-2"><bdi>978-962-209-026-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Set+Theory%2C+Part+I%2FII&rft.pages=27&rft.pub=Hong+Kong+University+Press&rft.date=1992-07-01&rft.isbn=978-962-209-026-2&rft.au=K.T.+Leung&rft.au=Doris+Lai-chue+Chen&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dcdmy2eOhJdkC%26pg%3DPA27&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAggarwal2021" class="citation book cs1">Aggarwal, M.L. (2021). "1. Sets". Understanding ISC Mathematics Class XI. Vol. 1. 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American Mathematical Soc. p. 30. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-3835-8" title="Special:BookSources/978-0-8218-3835-8"><bdi>978-0-8218-3835-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Set+Theory&rft.pages=30&rft.pub=American+Mathematical+Soc.&rft.date=2005&rft.isbn=978-0-8218-3835-8&rft.au=Felix+Hausdorff&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DyvVIdH16k0YC%26pg%3DPA30&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-Comninos2010-32"><a href="#cite_ref-Comninos2010_32-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeter_Comninos2010" class="citation book cs1">Peter Comninos (6 April 2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Kdb7-YnnOVwC&pg=PA7">Mathematical and Computer Programming Techniques for Computer Graphics</a>. Springer Science & Business Media. p. 7. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-84628-292-8" title="Special:BookSources/978-1-84628-292-8"><bdi>978-1-84628-292-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+and+Computer+Programming+Techniques+for+Computer+Graphics&rft.pages=7&rft.pub=Springer+Science+%26+Business+Media&rft.date=2010-04-06&rft.isbn=978-1-84628-292-8&rft.au=Peter+Comninos&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DKdb7-YnnOVwC%26pg%3DPA7&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-FOOTNOTEHalmos1960[httpsarchiveorgdetailsnaivesettheory00halmpage2mode2up_3]-33">^ <a href="#cite_ref-FOOTNOTEHalmos1960%5Bhttpsarchiveorgdetailsnaivesettheory00halmpage2mode2up_3%5D_33-0">a</a> <a href="#cite_ref-FOOTNOTEHalmos1960%5Bhttpsarchiveorgdetailsnaivesettheory00halmpage2mode2up_3%5D_33-1">b</a> <a href="#CITEREFHalmos1960">Halmos 1960</a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/naivesettheory00halm/page/2/mode/2up">3</a>. </li> <li id="cite_note-Tourlakis2003-34">^ <a href="#cite_ref-Tourlakis2003_34-0">a</a> <a href="#cite_ref-Tourlakis2003_34-1">b</a> <a href="#cite_ref-Tourlakis2003_34-2">c</a> <a href="#cite_ref-Tourlakis2003_34-3">d</a> <a href="#cite_ref-Tourlakis2003_34-4">e</a> <a href="#cite_ref-Tourlakis2003_34-5">f</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGeorge_Tourlakis2003" class="citation book cs1">George Tourlakis (13 February 2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nparMXao59QC&pg=PA137">Lectures in Logic and Set Theory: Volume 2, Set Theory</a>. Cambridge University Press. p. 137. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-139-43943-5" title="Special:BookSources/978-1-139-43943-5"><bdi>978-1-139-43943-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+in+Logic+and+Set+Theory%3A+Volume+2%2C+Set+Theory&rft.pages=137&rft.pub=Cambridge+University+Press&rft.date=2003-02-13&rft.isbn=978-1-139-43943-5&rft.au=George+Tourlakis&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DnparMXao59QC%26pg%3DPA137&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-Moschovakis1994-35"><a href="#cite_ref-Moschovakis1994_35-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYiannis_N._Moschovakis1994" class="citation book cs1">Yiannis N. 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Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-94180-4" title="Special:BookSources/978-3-540-94180-4"><bdi>978-3-540-94180-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Notes+on+Set+Theory&rft.pub=Springer+Science+%26+Business+Media&rft.date=1994&rft.isbn=978-3-540-94180-4&rft.au=Yiannis+N.+Moschovakis&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dndx0_6VCypcC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-Fleck2001-36"><a href="#cite_ref-Fleck2001_36-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArthur_Charles_Fleck2001" class="citation book cs1">Arthur Charles Fleck (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=c42oYf4zBzMC&pg=PA3">Formal Models of Computation: The Ultimate Limits of Computing</a>. World Scientific. p. 3. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-02-4500-9" title="Special:BookSources/978-981-02-4500-9"><bdi>978-981-02-4500-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Formal+Models+of+Computation%3A+The+Ultimate+Limits+of+Computing&rft.pages=3&rft.pub=World+Scientific&rft.date=2001&rft.isbn=978-981-02-4500-9&rft.au=Arthur+Charles+Fleck&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dc42oYf4zBzMC%26pg%3DPA3&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-Johnston2015-37"><a href="#cite_ref-Johnston2015_37-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilliam_Johnston2015" class="citation book cs1">William Johnston (25 September 2015). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=v4ueCgAAQBAJ&pg=PA7">The Lebesgue Integral for Undergraduates</a>. The Mathematical Association of America. p. 7. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-939512-07-9" title="Special:BookSources/978-1-939512-07-9"><bdi>978-1-939512-07-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Lebesgue+Integral+for+Undergraduates&rft.pages=7&rft.pub=The+Mathematical+Association+of+America&rft.date=2015-09-25&rft.isbn=978-1-939512-07-9&rft.au=William+Johnston&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dv4ueCgAAQBAJ%26pg%3DPA7&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-Smith2008-38"><a href="#cite_ref-Smith2008_38-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKarl_J._Smith2008" class="citation book cs1">Karl J. Smith (7 January 2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-0x2JszrkooC&pg=PA401">Mathematics: Its Power and Utility</a>. Cengage Learning. p. 401. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-495-38913-2" title="Special:BookSources/978-0-495-38913-2"><bdi>978-0-495-38913-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics%3A+Its+Power+and+Utility&rft.pages=401&rft.pub=Cengage+Learning&rft.date=2008-01-07&rft.isbn=978-0-495-38913-2&rft.au=Karl+J.+Smith&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D-0x2JszrkooC%26pg%3DPA401&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-Stillwell2013-39"><a href="#cite_ref-Stillwell2013_39-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohn_Stillwell2013" class="citation book cs1">John Stillwell (16 October 2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VPe8BAAAQBAJ">The Real Numbers: An Introduction to Set Theory and Analysis</a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-01577-4" title="Special:BookSources/978-3-319-01577-4"><bdi>978-3-319-01577-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Real+Numbers%3A+An+Introduction+to+Set+Theory+and+Analysis&rft.pub=Springer+Science+%26+Business+Media&rft.date=2013-10-16&rft.isbn=978-3-319-01577-4&rft.au=John+Stillwell&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DVPe8BAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-Tall2006-40"><a href="#cite_ref-Tall2006_40-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavid_Tall2006" class="citation book cs1">David Tall (11 April 2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=czKqBgAAQBAJ&pg=PA212">Advanced Mathematical Thinking</a>. Springer Science & Business Media. p. 211. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-306-47203-9" title="Special:BookSources/978-0-306-47203-9"><bdi>978-0-306-47203-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+Mathematical+Thinking&rft.pages=211&rft.pub=Springer+Science+%26+Business+Media&rft.date=2006-04-11&rft.isbn=978-0-306-47203-9&rft.au=David+Tall&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DczKqBgAAQBAJ%26pg%3DPA212&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-Cantor1878-41"><a href="#cite_ref-Cantor1878_41-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCantor1878" class="citation journal cs1">Cantor, Georg (1878). <a rel="nofollow" class="external text" href="http://www.digizeitschriften.de/dms/img/?PPN=PPN243919689_0084&DMDID=dmdlog15">"Ein Beitrag zur Mannigfaltigkeitslehre"</a>. <a href="/wiki/Journal_f%C3%BCr_die_Reine_und_Angewandte_Mathematik" class="mw-redirect" title="Journal für die Reine und Angewandte Mathematik">Journal für die Reine und Angewandte Mathematik</a>. 1878 (84): 242–258. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2Fcrll.1878.84.242">10.1515/crll.1878.84.242</a> (inactive 1 November 2024).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+f%C3%BCr+die+Reine+und+Angewandte+Mathematik&rft.atitle=Ein+Beitrag+zur+Mannigfaltigkeitslehre&rft.volume=1878&rft.issue=84&rft.pages=242-258&rft.date=1878&rft_id=info%3Adoi%2F10.1515%2Fcrll.1878.84.242&rft.aulast=Cantor&rft.aufirst=Georg&rft_id=http%3A%2F%2Fwww.digizeitschriften.de%2Fdms%2Fimg%2F%3FPPN%3DPPN243919689_0084%26DMDID%3Ddmdlog15&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: CS1 maint: DOI inactive as of November 2024 (<a href="/wiki/Category:CS1_maint:_DOI_inactive_as_of_November_2024" title="Category:CS1 maint: DOI inactive as of November 2024">link</a>) </li> <li id="cite_note-Cohen1963-42"><a href="#cite_ref-Cohen1963_42-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCohen1963" class="citation journal cs1">Cohen, Paul J. (December 15, 1963). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC221287">"The Independence of the Continuum Hypothesis"</a>. Proceedings of the National Academy of Sciences of the United States of America. 50 (6): 1143–1148. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1963PNAS...50.1143C">1963PNAS...50.1143C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.50.6.1143">10.1073/pnas.50.6.1143</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/71858">71858</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC221287">221287</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16578557">16578557</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+National+Academy+of+Sciences+of+the+United+States+of+America&rft.atitle=The+Independence+of+the+Continuum+Hypothesis&rft.volume=50&rft.issue=6&rft.pages=1143-1148&rft.date=1963-12-15&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC221287%23id-name%3DPMC&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F71858%23id-name%3DJSTOR&rft_id=info%3Abibcode%2F1963PNAS...50.1143C&rft_id=info%3Apmid%2F16578557&rft_id=info%3Adoi%2F10.1073%2Fpnas.50.6.1143&rft.aulast=Cohen&rft.aufirst=Paul+J.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC221287&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-FOOTNOTEHalmos1960[httpsarchiveorgdetailsnaivesettheory00halmpage18mode2up_19]-43"><a href="#cite_ref-FOOTNOTEHalmos1960%5Bhttpsarchiveorgdetailsnaivesettheory00halmpage18mode2up_19%5D_43-0">^</a> <a href="#CITEREFHalmos1960">Halmos 1960</a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/naivesettheory00halm/page/18/mode/2up">19</a>. </li> <li id="cite_note-FOOTNOTEHalmos1960[httpsarchiveorgdetailsnaivesettheory00halmpage20mode2up_20]-44"><a href="#cite_ref-FOOTNOTEHalmos1960%5Bhttpsarchiveorgdetailsnaivesettheory00halmpage20mode2up_20%5D_44-0">^</a> <a href="#CITEREFHalmos1960">Halmos 1960</a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/naivesettheory00halm/page/20/mode/2up">20</a>. </li> <li id="cite_note-BurgerStarbird2004-45"><a href="#cite_ref-BurgerStarbird2004_45-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEdward_B._BurgerMichael_Starbird2004" class="citation book cs1">Edward B. Burger; Michael Starbird (18 August 2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=M-qK8anbZmwC&pg=PA183">The Heart of Mathematics: An invitation to effective thinking</a>. Springer Science & Business Media. p. 183. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-931914-41-3" title="Special:BookSources/978-1-931914-41-3"><bdi>978-1-931914-41-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Heart+of+Mathematics%3A+An+invitation+to+effective+thinking&rft.pages=183&rft.pub=Springer+Science+%26+Business+Media&rft.date=2004-08-18&rft.isbn=978-1-931914-41-3&rft.au=Edward+B.+Burger&rft.au=Michael+Starbird&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DM-qK8anbZmwC%26pg%3DPA183&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-Mansour2012-46"><a href="#cite_ref-Mansour2012_46-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFToufik_Mansour2012" class="citation book cs1">Toufik Mansour (27 July 2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5NvrH4w8WGsC">Combinatorics of Set Partitions</a>. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4398-6333-6" title="Special:BookSources/978-1-4398-6333-6"><bdi>978-1-4398-6333-6</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+Set+Partitions&rft.pub=CRC+Press&rft.date=2012-07-27&rft.isbn=978-1-4398-6333-6&rft.au=Toufik+Mansour&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D5NvrH4w8WGsC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-FOOTNOTEHalmos1960[httpsarchiveorgdetailsnaivesettheory00halmpage28mode2up_28]-47"><a href="#cite_ref-FOOTNOTEHalmos1960%5Bhttpsarchiveorgdetailsnaivesettheory00halmpage28mode2up_28%5D_47-0">^</a> <a href="#CITEREFHalmos1960">Halmos 1960</a>, p. <a rel="nofollow" class="external text" href="https://archive.org/details/naivesettheory00halm/page/28/mode/2up">28</a>. </li> <li id="cite_note-Ferreirós2007-48"><a href="#cite_ref-Ferreir%C3%B3s2007_48-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJosé_Ferreirós2007" class="citation book cs1">José Ferreirós (16 August 2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TXRBwwEACAAJ">Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics</a>. Birkhäuser Basel. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-7643-8349-7" title="Special:BookSources/978-3-7643-8349-7"><bdi>978-3-7643-8349-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Labyrinth+of+Thought%3A+A+History+of+Set+Theory+and+Its+Role+in+Modern+Mathematics&rft.pub=Birkh%C3%A4user+Basel&rft.date=2007-08-16&rft.isbn=978-3-7643-8349-7&rft.au=Jos%C3%A9+Ferreir%C3%B3s&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DTXRBwwEACAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-Russ2004-49"><a href="#cite_ref-Russ2004_49-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteve_Russ2004" class="citation book cs1">Steve Russ (9 December 2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zp7cLQn0x3gC&pg=PR28">The Mathematical Works of Bernard Bolzano</a>. OUP Oxford. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-151370-1" title="Special:BookSources/978-0-19-151370-1"><bdi>978-0-19-151370-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Mathematical+Works+of+Bernard+Bolzano&rft.pub=OUP+Oxford&rft.date=2004-12-09&rft.isbn=978-0-19-151370-1&rft.au=Steve+Russ&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dzp7cLQn0x3gC%26pg%3DPR28&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-EwaldEwald1996-50"><a href="#cite_ref-EwaldEwald1996_50-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilliam_EwaldWilliam_Bragg_Ewald1996" class="citation book cs1">William Ewald; William Bragg Ewald (1996). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rykSDAAAQBAJ&pg=PA249">From Kant to Hilbert Volume 1: A Source Book in the Foundations of Mathematics</a>. OUP Oxford. p. 249. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-850535-8" title="Special:BookSources/978-0-19-850535-8"><bdi>978-0-19-850535-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=From+Kant+to+Hilbert+Volume+1%3A+A+Source+Book+in+the+Foundations+of+Mathematics&rft.pages=249&rft.pub=OUP+Oxford&rft.date=1996&rft.isbn=978-0-19-850535-8&rft.au=William+Ewald&rft.au=William+Bragg+Ewald&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DrykSDAAAQBAJ%26pg%3DPA249&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-RusnockSebestík2019-51"><a href="#cite_ref-RusnockSebest%C3%ADk2019_51-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPaul_RusnockJan_Sebestík2019" class="citation book cs1">Paul Rusnock; Jan Sebestík (25 April 2019). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-hqJDwAAQBAJ&pg=PA430">Bernard Bolzano: His Life and Work</a>. OUP Oxford. p. 430. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-255683-7" title="Special:BookSources/978-0-19-255683-7"><bdi>978-0-19-255683-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Bernard+Bolzano%3A+His+Life+and+Work&rft.pages=430&rft.pub=OUP+Oxford&rft.date=2019-04-25&rft.isbn=978-0-19-255683-7&rft.au=Paul+Rusnock&rft.au=Jan+Sebest%C3%ADk&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D-hqJDwAAQBAJ%26pg%3DPA430&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-Cantor.1895-52"><a href="#cite_ref-Cantor.1895_52-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGeorg_Cantor1895" class="citation journal cs1 cs1-prop-foreign-lang-source">Georg Cantor (Nov 1895). <a rel="nofollow" class="external text" href="http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN00225557X">"Beiträge zur Begründung der transfiniten Mengenlehre (1)"</a>. Mathematische Annalen (in German). 46 (4): 481–512.</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+%281%29&rft.volume=46&rft.issue=4&rft.pages=481-512&rft.date=1895-11&rft.au=Georg+Cantor&rft_id=http%3A%2F%2Fwww.digizeitschriften.de%2Fdms%2Fresolveppn%2F%3FPID%3DGDZPPN00225557X&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-53"><a href="#cite_ref-53">^</a> <a href="/wiki/Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a> (1903) <a href="/wiki/The_Principles_of_Mathematics" title="The Principles of Mathematics">The Principles of Mathematics</a>, chapter VI: Classes </li> <li id="cite_note-Ferreiros2001-54"><a href="#cite_ref-Ferreiros2001_54-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJose_Ferreiros2001" class="citation book cs1">Jose Ferreiros (1 November 2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DITy0nsYQQoC">Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics</a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-7643-5749-8" title="Special:BookSources/978-3-7643-5749-8"><bdi>978-3-7643-5749-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Labyrinth+of+Thought%3A+A+History+of+Set+Theory+and+Its+Role+in+Modern+Mathematics&rft.pub=Springer+Science+%26+Business+Media&rft.date=2001-11-01&rft.isbn=978-3-7643-5749-8&rft.au=Jose+Ferreiros&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DDITy0nsYQQoC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-55"><a href="#cite_ref-55">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRaatikainen2022" class="citation web cs1">Raatikainen, Panu (2022). Zalta, Edward N. (ed.). <a rel="nofollow" class="external text" href="https://plato.stanford.edu/archives/spr2022/entries/goedel-incompleteness/">"Gödel's Incompleteness Theorems"</a>. Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. Retrieved 2024-06-03.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Stanford+Encyclopedia+of+Philosophy&rft.atitle=G%C3%B6del%27s+Incompleteness+Theorems&rft.date=2022&rft.aulast=Raatikainen&rft.aufirst=Panu&rft_id=https%3A%2F%2Fplato.stanford.edu%2Farchives%2Fspr2022%2Fentries%2Fgoedel-incompleteness%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"> </li> </ol></div></div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(18)"><h2 id="References">References</h2> <a role="button" href="/w/index.php?title=Set_(mathematics)&action=edit&section=27" 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-18 collapsible-block" id="mf-section-18"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDauben1979" class="citation book cs1"><a href="/wiki/Joseph_Dauben" title="Joseph Dauben">Dauben, Joseph W.</a> (1979). <a rel="nofollow" class="external text" href="https://archive.org/details/georgcantorhisma0000daub">Georg Cantor: His Mathematics and Philosophy of the Infinite</a>. Boston: <a href="/wiki/Harvard_University_Press" title="Harvard University Press">Harvard University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-02447-2" title="Special:BookSources/0-691-02447-2"><bdi>0-691-02447-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Georg+Cantor%3A+His+Mathematics+and+Philosophy+of+the+Infinite&rft.place=Boston&rft.pub=Harvard+University+Press&rft.date=1979&rft.isbn=0-691-02447-2&rft.aulast=Dauben&rft.aufirst=Joseph+W.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgeorgcantorhisma0000daub&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHalmos1960" class="citation book cs1"><a href="/wiki/Paul_Halmos" title="Paul Halmos">Halmos, Paul R.</a> (1960). <a rel="nofollow" class="external text" href="https://archive.org/details/naivesettheory00halm">Naive Set Theory</a>. Princeton, N.J.: Van Nostrand. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-90092-6" title="Special:BookSources/0-387-90092-6"><bdi>0-387-90092-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Naive+Set+Theory&rft.place=Princeton%2C+N.J.&rft.pub=Van+Nostrand&rft.date=1960&rft.isbn=0-387-90092-6&rft.aulast=Halmos&rft.aufirst=Paul+R.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fnaivesettheory00halm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStoll1979" class="citation book cs1">Stoll, Robert R. (1979). Set Theory and Logic. Mineola, N.Y.: <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-63829-4" title="Special:BookSources/0-486-63829-4"><bdi>0-486-63829-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Set+Theory+and+Logic&rft.place=Mineola%2C+N.Y.&rft.pub=Dover+Publications&rft.date=1979&rft.isbn=0-486-63829-4&rft.aulast=Stoll&rft.aufirst=Robert+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVelleman2006" class="citation book cs1">Velleman, Daniel (2006). How To Prove It: A Structured Approach. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-67599-5" title="Special:BookSources/0-521-67599-5"><bdi>0-521-67599-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=How+To+Prove+It%3A+A+Structured+Approach&rft.pub=Cambridge+University+Press&rft.date=2006&rft.isbn=0-521-67599-5&rft.aulast=Velleman&rft.aufirst=Daniel&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASet+%28mathematics%29" class="Z3988"></li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(19)"><h2 id="External_links">External links</h2> <a role="button" href="/w/index.php?title=Set_(mathematics)&action=edit&section=28" title="Edit section: External links" 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-19 collapsible-block" id="mf-section-19"> <ul><li><a href="/wiki/File:Wiktionary-logo-en-v2.svg" class="mw-file-description"><noscript><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/16px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="16" height="16" class="mw-file-element" data-file-width="512" data-file-height="512"></noscript><span class="lazy-image-placeholder" style="width: 16px;height: 16px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/16px-Wiktionary-logo-en-v2.svg.png" data-alt="" data-width="16" data-height="16" data-srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/24px-Wiktionary-logo-en-v2.svg.png 1.5x, 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class="post-content footer-content"> <div id='mw-data-after-content'> <div class="read-more-container"></div> </div> <div id="p-lang"> <h4>Languages</h4> <section> <ul id="p-variants" class="minerva-languages"></ul> <ul class="minerva-languages"><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Menge_(Mathematik)" title="Menge (Mathematik) – Alemannic" lang="gsw" hreflang="gsw" data-title="Menge (Mathematik)" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target">Alemannisch</a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%B5%E1%89%A5%E1%88%B5%E1%89%A5" title="ስብስብ – Amharic" lang="am" hreflang="am" data-title="ስብስብ" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target">አማርኛ</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AC%D9%85%D9%88%D8%B9%D8%A9_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="مجموعة (رياضيات) – Arabic" lang="ar" hreflang="ar" data-title="مجموعة (رياضيات)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Conxuntu" title="Conxuntu – Asturian" lang="ast" hreflang="ast" data-title="Conxuntu" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C3%87oxluqlar" title="Çoxluqlar – Azerbaijani" lang="az" hreflang="az" data-title="Çoxluqlar" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A7%87%E0%A6%9F" title="সেট – Bangla" lang="bn" hreflang="bn" data-title="সেট" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Chi%CC%8Dp-ha%CC%8Dp" title="Chi̍p-ha̍p – Minnan" lang="nan" hreflang="nan" data-title="Chi̍p-ha̍p" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9A%D2%AF%D0%BC%D3%99%D0%BA%D0%BB%D0%B5%D0%BA" title="Күмәклек – Bashkir" lang="ba" hreflang="ba" data-title="Күмәклек" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target">Башҡортса</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D1%81%D1%82%D0%B2%D0%B0" title="Мноства – Belarusian" lang="be" hreflang="be" data-title="Мноства" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D1%81%D1%82%D0%B2%D0%B0" title="Мноства – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Мноства" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target">Беларуская (тарашкевіца)</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%BE" title="Множество – Bulgarian" lang="bg" hreflang="bg" data-title="Множество" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Skup_(matematika)" title="Skup (matematika) – Bosnian" lang="bs" hreflang="bs" data-title="Skup (matematika)" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Conjunt" title="Conjunt – Catalan" lang="ca" hreflang="ca" data-title="Conjunt" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%99%D1%8B%D1%88" title="Йыш – Chuvash" lang="cv" hreflang="cv" data-title="Йыш" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Mno%C5%BEina" title="Množina – Czech" lang="cs" hreflang="cs" data-title="Množina" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Set_(mathemateg)" title="Set (mathemateg) – Welsh" lang="cy" hreflang="cy" data-title="Set (mathemateg)" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/M%C3%A6ngde" title="Mængde – Danish" lang="da" hreflang="da" data-title="Mængde" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Menge_(Mathematik)" title="Menge (Mathematik) – German" lang="de" hreflang="de" data-title="Menge (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Hulk" title="Hulk – Estonian" lang="et" hreflang="et" data-title="Hulk" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%8D%CE%BD%CE%BF%CE%BB%CE%BF" title="Σύνολο – Greek" lang="el" hreflang="el" data-title="Σύνολο" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-myv mw-list-item"><a href="https://myv.wikipedia.org/wiki/%D0%92%D0%B5%D0%B9%D1%81%D1%81%D0%B0%D0%B5%D0%B2%D0%BA%D1%81" title="Вейссаевкс – Erzya" lang="myv" hreflang="myv" data-title="Вейссаевкс" data-language-autonym="Эрзянь" data-language-local-name="Erzya" class="interlanguage-link-target">Эрзянь</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Conjunto" title="Conjunto – Spanish" lang="es" hreflang="es" data-title="Conjunto" 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://eo.wikipedia.org/wiki/Aro_(matematiko)" title="Aro (matematiko) – Esperanto" lang="eo" hreflang="eo" data-title="Aro (matematiko)" 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://eu.wikipedia.org/wiki/Multzo" title="Multzo – Basque" lang="eu" hreflang="eu" data-title="Multzo" 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://fa.wikipedia.org/wiki/%D9%85%D8%AC%D9%85%D9%88%D8%B9%D9%87_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA)" title="مجموعه (ریاضیات) – Persian" lang="fa" hreflang="fa" data-title="مجموعه (ریاضیات)" 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://fr.wikipedia.org/wiki/Ensemble" title="Ensemble – French" lang="fr" hreflang="fr" data-title="Ensemble" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-fur mw-list-item"><a href="https://fur.wikipedia.org/wiki/Insiemi" title="Insiemi – Friulian" lang="fur" hreflang="fur" data-title="Insiemi" data-language-autonym="Furlan" data-language-local-name="Friulian" class="interlanguage-link-target">Furlan</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Tacar" title="Tacar – Irish" lang="ga" hreflang="ga" data-title="Tacar" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gd mw-list-item"><a href="https://gd.wikipedia.org/wiki/Seata" title="Seata – Scottish Gaelic" lang="gd" hreflang="gd" data-title="Seata" data-language-autonym="Gàidhlig" data-language-local-name="Scottish Gaelic" class="interlanguage-link-target">Gàidhlig</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Conxunto" title="Conxunto – Galician" lang="gl" hreflang="gl" data-title="Conxunto" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E9%9B%86%E5%90%88" title="集合 – Gan" lang="gan" hreflang="gan" data-title="集合" data-language-autonym="贛語" data-language-local-name="Gan" class="interlanguage-link-target">贛語</a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%9E%D0%BB%D0%BD" title="Олн – Kalmyk" lang="xal" hreflang="xal" data-title="Олн" data-language-autonym="Хальмг" data-language-local-name="Kalmyk" class="interlanguage-link-target">Хальмг</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A7%91%ED%95%A9" title="집합 – Korean" lang="ko" hreflang="ko" data-title="집합" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B2%D5%A1%D5%A6%D5%B4%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Բազմություն – Armenian" lang="hy" hreflang="hy" data-title="Բազմություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%AE%E0%A5%81%E0%A4%9A%E0%A5%8D%E0%A4%9A%E0%A4%AF_(%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4)" title="समुच्चय (गणित) – Hindi" lang="hi" hreflang="hi" data-title="समुच्चय (गणित)" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Skup" title="Skup – Croatian" lang="hr" hreflang="hr" data-title="Skup" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Ensemblo" title="Ensemblo – Ido" lang="io" hreflang="io" data-title="Ensemblo" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target">Ido</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Himpunan_(matematika)" title="Himpunan (matematika) – Indonesian" lang="id" hreflang="id" data-title="Himpunan (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Insimul" title="Insimul – Interlingua" lang="ia" hreflang="ia" data-title="Insimul" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Mengi" title="Mengi – Icelandic" lang="is" hreflang="is" data-title="Mengi" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Insieme" title="Insieme – Italian" lang="it" hreflang="it" data-title="Insieme" 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://he.wikipedia.org/wiki/%D7%A7%D7%91%D7%95%D7%A6%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="קבוצה (מתמטיקה) – Hebrew" lang="he" hreflang="he" data-title="קבוצה (מתמטיקה)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%97%E0%B2%A3_(%E0%B2%97%E0%B2%A3%E0%B2%BF%E0%B2%A4)" title="ಗಣ (ಗಣಿತ) – Kannada" lang="kn" hreflang="kn" data-title="ಗಣ (ಗಣಿತ)" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target">ಕನ್ನಡ</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A1%E1%83%98%E1%83%9B%E1%83%A0%E1%83%90%E1%83%95%E1%83%9A%E1%83%94" title="სიმრავლე – Georgian" lang="ka" hreflang="ka" data-title="სიმრავლე" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%91%D3%A9%D0%BB%D1%96%D0%BA" title="Бөлік – Kazakh" lang="kk" hreflang="kk" data-title="Бөлік" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Ansanm" title="Ansanm – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Ansanm" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target">Kriyòl gwiyannen</a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Kom" title="Kom – Kurdish" lang="ku" hreflang="ku" data-title="Kom" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target">Kurdî</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Copia" title="Copia – Latin" lang="la" hreflang="la" data-title="Copia" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Kopa" title="Kopa – Latvian" lang="lv" hreflang="lv" data-title="Kopa" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Aib%C4%97" title="Aibė – Lithuanian" lang="lt" hreflang="lt" data-title="Aibė" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-ln mw-list-item"><a href="https://ln.wikipedia.org/wiki/El%C9%94ng%C9%94%CC%81t%C9%9B%CC%82_lisang%C3%A1" title="Elɔngɔ́tɛ̂ lisangá – Lingala" lang="ln" hreflang="ln" data-title="Elɔngɔ́tɛ̂ lisangá" data-language-autonym="Lingála" data-language-local-name="Lingala" class="interlanguage-link-target">Lingála</a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Insemma" title="Insemma – Lombard" lang="lmo" hreflang="lmo" data-title="Insemma" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target">Lombard</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Halmaz_(matematika)" title="Halmaz (matematika) – Hungarian" lang="hu" hreflang="hu" data-title="Halmaz (matematika)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%BE" title="Множество – Macedonian" lang="mk" hreflang="mk" data-title="Множество" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%97%E0%B4%A3%E0%B4%82_(%E0%B4%97%E0%B4%A3%E0%B4%BF%E0%B4%A4%E0%B4%82)" title="ഗണം (ഗണിതം) – Malayalam" lang="ml" hreflang="ml" data-title="ഗണം (ഗണിതം)" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-xmf mw-list-item"><a href="https://xmf.wikipedia.org/wiki/%E1%83%9B%E1%83%98%E1%83%90%E1%83%A0%E1%83%94" title="მიარე – Mingrelian" lang="xmf" hreflang="xmf" data-title="მიარე" data-language-autonym="მარგალური" data-language-local-name="Mingrelian" class="interlanguage-link-target">მარგალური</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Set" title="Set – Malay" lang="ms" hreflang="ms" data-title="Set" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%9E%D0%BB%D0%BE%D0%BD%D0%BB%D0%BE%D0%B3" title="Олонлог – Mongolian" lang="mn" hreflang="mn" data-title="Олонлог" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target">Монгол</a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%A1%E1%80%85%E1%80%AF" title="အစု – Burmese" lang="my" hreflang="my" data-title="အစု" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target">မြန်မာဘာသာ</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Verzameling_(wiskunde)" title="Verzameling (wiskunde) – Dutch" lang="nl" hreflang="nl" data-title="Verzameling (wiskunde)" 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://ja.wikipedia.org/wiki/%E9%9B%86%E5%90%88" title="集合 – Japanese" lang="ja" hreflang="ja" data-title="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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Mengde" title="Mengde – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Mengde" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Mengd" title="Mengd – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Mengd" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-nov mw-list-item"><a href="https://nov.wikipedia.org/wiki/Ensemble" title="Ensemble – Novial" lang="nov" hreflang="nov" data-title="Ensemble" data-language-autonym="Novial" data-language-local-name="Novial" class="interlanguage-link-target">Novial</a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Ensemble" title="Ensemble – Occitan" lang="oc" hreflang="oc" data-title="Ensemble" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/To%CA%BBplam_(matematika)" title="Toʻplam (matematika) – Uzbek" lang="uz" hreflang="uz" data-title="Toʻplam (matematika)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B8%E0%A9%88%E0%A9%B1%E0%A8%9F_(%E0%A8%97%E0%A8%A3%E0%A8%BF%E0%A8%A4)" title="ਸੈੱਟ (ਗਣਿਤ) – Punjabi" lang="pa" hreflang="pa" data-title="ਸੈੱਟ (ਗਣਿਤ)" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-jam mw-list-item"><a href="https://jam.wikipedia.org/wiki/Set_(matimatix)" title="Set (matimatix) – Jamaican Creole English" lang="jam" hreflang="jam" data-title="Set (matimatix)" data-language-autonym="Patois" data-language-local-name="Jamaican Creole English" class="interlanguage-link-target">Patois</a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Ansem" title="Ansem – Piedmontese" lang="pms" hreflang="pms" data-title="Ansem" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target">Piemontèis</a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Koppel_(Mathematik)" title="Koppel (Mathematik) – Low German" lang="nds" hreflang="nds" data-title="Koppel (Mathematik)" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target">Plattdüütsch</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Zbi%C3%B3r" title="Zbiór – Polish" lang="pl" hreflang="pl" data-title="Zbiór" 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://pt.wikipedia.org/wiki/Conjunto" title="Conjunto – Portuguese" lang="pt" hreflang="pt" data-title="Conjunto" 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://ro.wikipedia.org/wiki/Mul%C8%9Bime" title="Mulțime – Romanian" lang="ro" hreflang="ro" data-title="Mulțime" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-qu mw-list-item"><a href="https://qu.wikipedia.org/wiki/Tantachisqa" title="Tantachisqa – Quechua" lang="qu" hreflang="qu" data-title="Tantachisqa" data-language-autonym="Runa Simi" data-language-local-name="Quechua" class="interlanguage-link-target">Runa Simi</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%BE" title="Множество – Russian" lang="ru" hreflang="ru" data-title="Множество" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Bashk%C3%ABsit%C3%AB" title="Bashkësitë – Albanian" lang="sq" hreflang="sq" data-title="Bashkësitë" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Nzemi" title="Nzemi – Sicilian" lang="scn" hreflang="scn" data-title="Nzemi" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target">Sicilianu</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Set" title="Set – Simple English" lang="en-simple" hreflang="en-simple" data-title="Set" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Mno%C5%BEina" title="Množina – Slovak" lang="sk" hreflang="sk" data-title="Množina" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Mno%C5%BEica" title="Množica – Slovenian" lang="sl" hreflang="sl" data-title="Množica" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Mynga_(matymatyka)" title="Mynga (matymatyka) – Silesian" lang="szl" hreflang="szl" data-title="Mynga (matymatyka)" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target">Ślůnski</a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Qaybta_(xisaab)" title="Qaybta (xisaab) – Somali" lang="so" hreflang="so" data-title="Qaybta (xisaab)" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target">Soomaaliga</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%A9%DB%86%D9%85%DB%95%DA%B5%DB%95_(%D9%85%D8%A7%D8%AA%D9%85%D8%A7%D8%AA%DB%8C%DA%A9)" title="کۆمەڵە (ماتماتیک) – Central Kurdish" lang="ckb" hreflang="ckb" data-title="کۆمەڵە (ماتماتیک)" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A1%D0%BA%D1%83%D0%BF" title="Скуп – Serbian" lang="sr" hreflang="sr" data-title="Скуп" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Skup" title="Skup – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Skup" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Joukko" title="Joukko – Finnish" lang="fi" hreflang="fi" data-title="Joukko" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/M%C3%A4ngd" title="Mängd – Swedish" lang="sv" hreflang="sv" data-title="Mängd" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Pangkat_(matematika)" title="Pangkat (matematika) – Tagalog" lang="tl" hreflang="tl" data-title="Pangkat (matematika)" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target">Tagalog</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AE%A3%E0%AE%AE%E0%AF%8D_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D)" title="கணம் (கணிதம்) – Tamil" lang="ta" hreflang="ta" data-title="கணம் (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%B8%E0%B0%AE%E0%B0%BF%E0%B0%A4%E0%B1%81%E0%B0%B2%E0%B1%81" title="సమితులు – Telugu" lang="te" hreflang="te" data-title="సమితులు" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target">తెలుగు</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%8B%E0%B8%95_(%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C)" title="เซต (คณิตศาสตร์) – Thai" lang="th" hreflang="th" data-title="เซต (คณิตศาสตร์)" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/K%C3%BCme" title="Küme – Turkish" lang="tr" hreflang="tr" data-title="Küme" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D0%B6%D0%B8%D0%BD%D0%B0" title="Множина – Ukrainian" lang="uk" hreflang="uk" data-title="Множина" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%B7%D8%A7%D9%82%D9%85_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C)" title="طاقم (ریاضی) – Urdu" lang="ur" hreflang="ur" data-title="طاقم (ریاضی)" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/T%E1%BA%ADp_h%E1%BB%A3p_(to%C3%A1n_h%E1%BB%8Dc)" title="Tập hợp (toán học) – Vietnamese" lang="vi" hreflang="vi" data-title="Tập hợp (toán học)" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Hulk" title="Hulk – Võro" lang="vro" hreflang="vro" data-title="Hulk" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target">Võro</a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E9%9B%86" title="集 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="集" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target">文言</a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/Verzoamelienge" title="Verzoamelienge – West Flemish" lang="vls" hreflang="vls" data-title="Verzoamelienge" data-language-autonym="West-Vlams" data-language-local-name="West Flemish" class="interlanguage-link-target">West-Vlams</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E9%9B%86%E5%90%88%EF%BC%88%E6%95%B0%E5%AD%A6%EF%BC%89" title="集合（数学） – Wu" lang="wuu" hreflang="wuu" data-title="集合（数学）" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%92%D7%A2%D7%96%D7%A2%D7%9E%D7%9C_(%D7%9E%D7%90%D7%98%D7%A2%D7%9E%D7%90%D7%98%D7%99%D7%A7)" title="געזעמל (מאטעמאטיק) – Yiddish" lang="yi" hreflang="yi" data-title="געזעמל (מאטעמאטיק)" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target">ייִדיש</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%9B%86%E5%90%88" title="集合 – Cantonese" lang="yue" hreflang="yue" data-title="集合" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%9B%86%E5%90%88_(%E6%95%B0%E5%AD%A6)" title="集合 (数学) – Chinese" lang="zh" hreflang="zh" data-title="集合 (数学)" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li></ul> </section> 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</div> <div class="mw-notification-area" data-mw="interface"></div>  <script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-7fd659d96c-sqhqj","wgBackendResponseTime":225,"wgPageParseReport":{"limitreport":{"cputime":"1.018","walltime":"1.757","ppvisitednodes":{"value":11365,"limit":1000000},"postexpandincludesize":{"value":217289,"limit":2097152},"templateargumentsize":{"value":14617,"limit":2097152},"expansiondepth":{"value":12,"limit":100},"expensivefunctioncount":{"value":16,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":227028,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 982.922 1 -total"," 29.07% 285.779 1 Template:Reflist"," 23.12% 227.209 41 Template:Cite_book"," 14.92% 146.611 121 Template:Math"," 11.04% 108.509 6 Template:Navbox"," 9.66% 94.933 1 Template:In_lang"," 7.64% 75.047 1 Template:Set_theory"," 6.90% 67.790 14 Template:Sfn"," 5.94% 58.412 1 Template:About"," 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