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class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Relationship between two sets, defined by a set of ordered pairs</div> <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 basic notions of (homogeneous binary) relations in mathematics. For a more in-depth treatment, see <a href="/wiki/Binary_relation" title="Binary relation">Binary relation</a>. For relations on any number of elements, see <a href="/wiki/Finitary_relation" title="Finitary relation">Finitary relation</a>.</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Relaci%C3%B3n_binaria_01.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Relaci%C3%B3n_binaria_01.svg/300px-Relaci%C3%B3n_binaria_01.svg.png" decoding="async" width="300" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Relaci%C3%B3n_binaria_01.svg/450px-Relaci%C3%B3n_binaria_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/68/Relaci%C3%B3n_binaria_01.svg/600px-Relaci%C3%B3n_binaria_01.svg.png 2x" data-file-width="250" data-file-height="200" /></a><figcaption>Illustration of an example relation on a set A = { a, b, c, d }. An arrow from x to y indicates that the relation holds between x and y. The relation is represented by the set { (a,a), (a,b), (a,d), (b,a), (b,d), (c,b), (d,c), (d,d) } of ordered pairs.</figcaption></figure> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a relation denotes some kind of relationship between two <a href="/wiki/Mathematical_object" title="Mathematical object">objects</a> in a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a>, which may or may not hold.<a href="#cite_note-1">[1]</a> As an example, "is less than" is a relation on the set of <a href="/wiki/Natural_number" title="Natural number">natural numbers</a>; it holds, for instance, between the values 1 and 3 (denoted as 1 < 3), and likewise between 3 and 4 (denoted as 3 < 4), but not between the values 3 and 1 nor between 4 and 4, that is, 3 < 1 and 4 < 4 both evaluate to false. As another example, "is sister of" is a relation on the set of all people, it holds e.g. between <a href="/wiki/Marie_Curie" title="Marie Curie">Marie Curie</a> and <a href="/wiki/Bronis%C5%82awa_D%C5%82uska" title="Bronisława Dłuska">Bronisława Dłuska</a>, and likewise vice versa. Set members may not be in relation "to a certain degree" – either they are in relation or they are not. Formally, a relation R over a set X can be seen as a set of <a href="/wiki/Ordered_pair" title="Ordered pair">ordered pairs</a> (x,y) of members of X.<a href="#cite_note-FOOTNOTECodd1970-2">[2]</a> The relation R holds between x and y if (x,y) is a member of R. For example, the relation "is less than" on the natural numbers is an infinite set Rless of pairs of natural numbers that contains both (1,3) and (3,4), but neither (3,1) nor (4,4). The relation "is a <a href="/wiki/Nontrivial_divisor" class="mw-redirect" title="Nontrivial divisor">nontrivial divisor</a> of" on the set of one-digit natural numbers is sufficiently small to be shown here: Rdv = { (2,4), (2,6), (2,8), (3,6), (3,9), (4,8) }; for example 2 is a nontrivial divisor of 8, but not vice versa, hence (2,8) ∈ Rdv, but (8,2) ∉ Rdv. If R is a relation that holds for x and y one often writes xRy. For most common relations in mathematics, special symbols are introduced, like "<" for "is less than", and "|" for "is a nontrivial divisor of", and, most popular "=" for "is equal to". For example, "1 < 3", "1 is less than 3", and "(1,3) ∈ Rless" mean all the same; some authors also write "(1,3) ∈ (<)". Various properties of relations are investigated. A relation R is reflexive if xRx holds for all x, and irreflexive if xRx holds for no x. It is symmetric if xRy always implies yRx, and asymmetric if xRy implies that yRx is impossible. It is transitive if xRy and yRz always implies xRz. For example, "is less than" is irreflexive, asymmetric, and transitive, but neither reflexive nor symmetric. "is sister of" is transitive, but neither reflexive (e.g. <a href="/wiki/Pierre_Curie" title="Pierre Curie">Pierre Curie</a> is not a sister of himself), nor symmetric, nor asymmetric; while being irreflexive or not may be a matter of definition (is every woman a sister of herself?), "is ancestor of" is transitive, while "is parent of" is not. Mathematical theorems are known about combinations of relation properties, such as "a transitive relation is irreflexive if, and only if, it is asymmetric". Of particular importance are relations that satisfy certain combinations of properties. A <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">partial order</a> is a relation that is reflexive, antisymmetric, and transitive,<a href="#cite_note-FOOTNOTEHalmos1968Ch_14-3">[3]</a> an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> is a relation that is reflexive, symmetric, and transitive,<a href="#cite_note-FOOTNOTEHalmos1968Ch_7-4">[4]</a> a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> is a relation that is right-unique and left-total (see <a class="mw-selflink-fragment" href="#Properties_of_relations">below</a>).<a href="#cite_note-5">[5]</a><a href="#cite_note-FOOTNOTEHalmos1968Ch_8-6">[6]</a> Since relations are sets, they can be manipulated using set operations, including <a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a>, <a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a>, and <a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complementation</a>, leading to the <a href="/wiki/Algebra_of_sets" title="Algebra of sets">algebra of sets</a>. Furthermore, the <a href="/wiki/Calculus_of_relations" class="mw-redirect" title="Calculus of relations">calculus of relations</a> includes the operations of taking the <a href="/wiki/Converse_relation" title="Converse relation">converse</a> and <a href="/wiki/Composition_of_relations" title="Composition of relations">composing relations</a>.<a href="#cite_note-Schroder.1895-7">[7]</a><a href="#cite_note-Lewis.1918-8">[8]</a><a href="#cite_note-FOOTNOTESchmidt2010Chapt._5-9">[9]</a> The above concept of relation<a href="#cite_note-10">[a]</a> has been generalized to admit relations between members of two different sets (<a href="/wiki/Heterogeneous_relation" class="mw-redirect" title="Heterogeneous relation">heterogeneous relation</a>, like "lies on" between the set of all <a href="/wiki/Point_(geometry)" title="Point (geometry)">points</a> and that of all <a href="/wiki/Line_(geometry)" title="Line (geometry)">lines</a> in geometry), relations between three or more sets (<a href="/wiki/Finitary_relation" title="Finitary relation">finitary relation</a>, like "person x lives in town y at time z"), and relations between <a href="/wiki/Class_(mathematics)" class="mw-redirect" title="Class (mathematics)">classes</a><a href="#cite_note-11">[b]</a> (like "is an element of" on the class of all sets, see <a href="/wiki/Binary_relation#Sets_versus_classes" title="Binary relation">Binary relation § Sets versus classes</a>). <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2>[<a href="/w/index.php?title=Relation_(mathematics)&action=edit&section=1" title="Edit section: Definition">edit</a>]</div> Given a set X, a relation R over X is a set of <a href="/wiki/Ordered_pair" title="Ordered pair">ordered pairs</a> of elements from X, formally: R ⊆ { (x,y) | x, y ∈ X }.<a href="#cite_note-FOOTNOTECodd1970-2">[2]</a><a href="#cite_note-FOOTNOTEEnderton1977Ch_3._p._40-12">[10]</a> The statement (x,y) ∈ R reads "x is R-related to y" and is written in <a href="/wiki/Infix_notation" title="Infix notation">infix notation</a> as xRy.<a href="#cite_note-Schroder.1895-7">[7]</a><a href="#cite_note-Lewis.1918-8">[8]</a> The order of the elements is important; if x ≠ y then yRx can be true or false independently of xRy. For example, 3 divides 9, but 9 does not divide 3. <div class="mw-heading mw-heading2"><h2 id="Representation_of_relations">Representation of relations</h2>[<a href="/w/index.php?title=Relation_(mathematics)&action=edit&section=2" title="Edit section: Representation of relations">edit</a>]</div> <table style="float: right"> <tbody><tr> <td><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Rotated_ellipse_bg_red.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Rotated_ellipse_bg_red.svg/220px-Rotated_ellipse_bg_red.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Rotated_ellipse_bg_red.svg/330px-Rotated_ellipse_bg_red.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Rotated_ellipse_bg_red.svg/440px-Rotated_ellipse_bg_red.svg.png 2x" data-file-width="320" data-file-height="320" /></a><figcaption>The representation of the relation Rel = { (x,y) ∈ R × R | x2 + xy + y2 = 1 } as a 2D-plot yields an <a href="/wiki/Ellipse" title="Ellipse">ellipse</a>.</figcaption></figure> </td></tr></tbody></table> <table class="wikitable" style="float: right"> <tbody><tr> <th style="background:#EAECF0;background:linear-gradient(to top right,#EAECF0 49%,#AAA 49.5%,#AAA 50.5%,#EAECF0 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right">y</div><div style="margin-right:2em;text-align:left">x</div> </th> <th>1</th> <th>2</th> <th>3</th> <th>4</th> <th>6</th> <th>12 </th></tr> <tr> <th>1 </th> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /></td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /></td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /></td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /></td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /></td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td></tr> <tr> <th>2 </th> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /></td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /></td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /></td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /></td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /></td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td></tr> <tr> <th>3 </th> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /></td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /></td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /></td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /></td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /></td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td></tr> <tr> <th>4 </th> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /></td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /></td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /></td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /></td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /></td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td></tr> <tr> <th>6 </th> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /></td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /></td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /></td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /></td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /></td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td></tr> <tr> <th>12 </th> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /></td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /></td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /></td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /></td> <td data-sort-value="No" style="background: #FFE3E3; color:black; vertical-align: middle; text-align: center;" class="table-no2"><img alt="No" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /></td> <td data-sort-value="Yes" style="background: #DFD; color:black; vertical-align: middle; text-align: center;" class="table-yes2"><img alt="Yes" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/13px-Check-green.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/20px-Check-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Check-green.svg/26px-Check-green.svg.png 2x" data-file-width="512" data-file-height="512" /> </td></tr> <tr> <td colspan="7">Representation of Rdiv as a Boolean matrix </td></tr></tbody></table> <table style="float: right"> <tbody><tr> <td><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Relation_repr_12div_svg.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Relation_repr_12div_svg.svg/220px-Relation_repr_12div_svg.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Relation_repr_12div_svg.svg/330px-Relation_repr_12div_svg.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Relation_repr_12div_svg.svg/440px-Relation_repr_12div_svg.svg.png 2x" data-file-width="248" data-file-height="248" /></a><figcaption>Representation Rdiv as <a href="/wiki/Hasse_diagram" title="Hasse diagram">Hasse diagram</a> (black lines) and <a href="/wiki/Directed_graph" title="Directed graph">directed graph</a> (all lines)</figcaption></figure> </td></tr></tbody></table> A relation R on a finite set X may be represented as: <ul><li><a href="/wiki/Directed_graph" title="Directed graph">Directed graph</a>: Each member of X corresponds to a vertex; a directed edge from x to y exists if and only if (x,y) ∈ R.</li> <li><a href="/wiki/Boolean_matrix" title="Boolean matrix">Boolean matrix</a>: The members of X are arranged in some fixed sequence x1, ..., xn; the matrix has dimensions n × n, with the element in line i, column j, being <a href="/wiki/File:Yes_check.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/20px-Yes_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/26px-Yes_check.svg.png 2x" data-file-width="600" data-file-height="600" /></a>, if (xi,xj) ∈ R, and <a href="/wiki/File:Dark_Red_x.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/13px-Dark_Red_x.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/20px-Dark_Red_x.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Dark_Red_x.svg/26px-Dark_Red_x.svg.png 2x" data-file-width="600" data-file-height="600" /></a>, otherwise.</li> <li><a href="/wiki/Plot_(graphics)" title="Plot (graphics)">2D-plot</a>: As a generalization of a Boolean matrix, a relation on the –infinite– set R of <a href="/wiki/Real_number" title="Real number">real numbers</a> can be represented as a two-dimensional geometric figure: using <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a>, draw a <a href="/wiki/Point_(geometry)" title="Point (geometry)">point</a> at (x,y) whenever (x,y) ∈ R.</li></ul> A transitive<a href="#cite_note-13">[c]</a> relation R on a finite set X may be also represented as <ul><li><a href="/wiki/Hasse_diagram" title="Hasse diagram">Hasse diagram</a>: Each member of X corresponds to a vertex; directed edges are drawn such that a <a href="/wiki/Directed_path" class="mw-redirect" title="Directed path">directed path</a> from x to y exists if and only if (x,y) ∈ R. Compared to a directed-graph representation, a Hasse diagram needs fewer edges, leading to a less tangled image. Since the relation "a directed path exists from x to y" is transitive, only transitive relations can be represented in Hasse diagrams. Usually the diagram is laid out such that all edges point in an upward direction, and the arrows are omitted.</li></ul> For example, on the set of all divisors of 12, define the relation Rdiv by <dl><dd>x Rdiv y if x is a divisor of y and x ≠ y.</dd></dl> Formally, X = { 1, 2, 3, 4, 6, 12 } and Rdiv = { (1,2), (1,3), (1,4), (1,6), (1,12), (2,4), (2,6), (2,12), (3,6), (3,12), (4,12), (6,12) }. The representation of Rdiv as a Boolean matrix is shown in the middle table; the representation both as a Hasse diagram and as a directed graph is shown in the left picture. The following are equivalent: <ul><li>x Rdiv y is true.</li> <li>(x,y) ∈ Rdiv.</li> <li>A path from x to y exists in the Hasse diagram representing Rdiv.</li> <li>An edge from x to y exists in the directed graph representing Rdiv.</li> <li>In the Boolean matrix representing Rdiv, the element in line x, column y is "<a href="/wiki/File:Yes_check.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/13px-Yes_check.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/20px-Yes_check.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/fb/Yes_check.svg/26px-Yes_check.svg.png 2x" data-file-width="600" data-file-height="600" /></a>".</li></ul> As another example, define the relation Rel on R by <dl><dd>x Rel y if x2 + xy + y2 = 1.</dd></dl> The representation of Rel as a 2D-plot obtains an ellipse, see right picture. Since R is not finite, neither a directed graph, nor a finite Boolean matrix, nor a Hasse diagram can be used to depict Rel. <div class="mw-heading mw-heading2"><h2 id="Properties_of_relations">Properties of relations</h2>[<a href="/w/index.php?title=Relation_(mathematics)&action=edit&section=3" title="Edit section: Properties of relations">edit</a>]</div> Some important properties that a relation R over a set X may have are: <dl><dt><a href="/wiki/Reflexive_relation" title="Reflexive relation">Reflexive</a></dt> <dd>for all x ∈ X, xRx. For example, ≥ is a reflexive relation but > is not.</dd></dl> <dl><dt><a href="/wiki/Irreflexive_relation" class="mw-redirect" title="Irreflexive relation">Irreflexive</a> (or strict)</dt> <dd>for all x ∈ X, not xRx. For example, > is an irreflexive relation, but ≥ is not.</dd></dl> The previous 2 alternatives are not exhaustive; e.g., the red relation y = x2 given in the diagram below is neither irreflexive, nor reflexive, since it contains the pair (0,0), but not (2,2), respectively. <dl><dt><a href="/wiki/Symmetric_relation" title="Symmetric relation">Symmetric</a></dt> <dd>for all x, y ∈ X, if xRy then yRx. For example, "is a blood relative of" is a symmetric relation, because x is a blood relative of y if and only if y is a blood relative of x.</dd></dl> <dl><dt><a href="/wiki/Antisymmetric_relation" title="Antisymmetric relation">Antisymmetric</a></dt> <dd>for all x, y ∈ X, if xRy and yRx then x = y. For example, ≥ is an antisymmetric relation; so is >, but <a href="/wiki/Vacuous_truth" title="Vacuous truth">vacuously</a> (the condition in the definition is always false).<a href="#cite_note-FOOTNOTESmithEggenSt._Andre2006160-14">[11]</a></dd></dl> <dl><dt><a href="/wiki/Asymmetric_relation" title="Asymmetric relation">Asymmetric</a></dt> <dd>for all x, y ∈ X, if xRy then not yRx. A relation is asymmetric if and only if it is both antisymmetric and irreflexive.<a href="#cite_note-FOOTNOTENievergelt2002[httpsbooksgooglecombooksid_H_nJdagqL8CpgPA158_158]-15">[12]</a> For example, > is an asymmetric relation, but ≥ is not.</dd></dl> Again, the previous 3 alternatives are far from being exhaustive; as an example over the natural numbers, the relation xRy defined by x > 2 is neither symmetric (e.g. 5R1, but not 1R5) nor antisymmetric (e.g. 6R4, but also 4R6), let alone asymmetric. <dl><dt><a href="/wiki/Transitive_relation" title="Transitive relation">Transitive</a></dt> <dd>for all x, y, z ∈ X, if xRy and yRz then xRz. A transitive relation is irreflexive if and only if it is asymmetric.<a href="#cite_note-FOOTNOTEFlaškaJežekKepkaKortelainen2007p.1_Lemma_1.1_(iv)._This_source_refers_to_asymmetric_relations_as_"strictly_antisymmetric".-16">[13]</a> For example, "is ancestor of" is a transitive relation, while "is parent of" is not.</dd></dl> <dl><dt><a href="/wiki/Connected_relation" title="Connected relation">Connected</a></dt> <dd>for all x, y ∈ X, if x ≠ y then xRy or yRx. For example, on the natural numbers, < is connected, while "is a divisor of" is not (e.g. neither 5R7 nor 7R5).</dd></dl> <dl><dt><a href="/wiki/Connected_relation" title="Connected relation">Strongly connected</a></dt> <dd>for all x, y ∈ X, xRy or yRx. For example, on the natural numbers, ≤ is strongly connected, but < is not. A relation is strongly connected if, and only if, it is connected and reflexive.</dd></dl> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:The_four_types_of_binary_relations.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/The_four_types_of_binary_relations.png/220px-The_four_types_of_binary_relations.png" decoding="async" width="220" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/The_four_types_of_binary_relations.png/330px-The_four_types_of_binary_relations.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b6/The_four_types_of_binary_relations.png/440px-The_four_types_of_binary_relations.png 2x" data-file-width="6865" data-file-height="6888" /></a><figcaption>Examples of four types of relations over the <a href="/wiki/Real_number" title="Real number">real numbers</a>: one-to-one (in green), one-to-many (in blue), many-to-one (in red), many-to-many (in black). 2D-plot representation is used.</figcaption></figure> Uniqueness properties: <dl><dt>Injective<a href="#cite_note-heterogeneous-17">[d]</a> (also called left-unique<a href="#cite_note-kkm-18">[14]</a>)</dt> <dd>For all x, y, z ∈ X, if xRy and zRy then x = z. For example, the green and blue relations in the diagram are injective, but the red one is not (as it relates both −1 and 1 to 1), nor is the black one (as it relates both −1 and 1 to 0).</dd> <dt>Functional<a href="#cite_note-19">[15]</a><a href="#cite_note-20">[16]</a><a href="#cite_note-21">[17]</a><a href="#cite_note-heterogeneous-17">[d]</a> (also called right-unique,<a href="#cite_note-kkm-18">[14]</a> right-definite<a href="#cite_note-FOOTNOTEMäs2007-22">[18]</a> or univalent<a href="#cite_note-FOOTNOTESchmidt2010Chapt._5-9">[9]</a>)</dt> <dd>For all x, y, z ∈ X, if xRy and xRz then y = z. Such a relation is called a <a href="/wiki/Partial_function" title="Partial function">partial function</a>. For example, the red and green relations in the diagram are functional, but the blue one is not (as it relates 1 to both −1 and 1), nor is the black one (as it relates 0 to both −1 and 1).</dd></dl> Totality properties: <dl><dt><a href="/wiki/Serial_relation" title="Serial relation">Serial</a><a href="#cite_note-heterogeneous-17">[d]</a> (also called total or left-total)</dt> <dd>For all x ∈ X, there exists some y ∈ X such that xRy. Such a relation is called a <a href="/wiki/Multivalued_function" title="Multivalued function">multivalued function</a>. For example, the red and green relations in the diagram are total, but the blue one is not (as it does not relate −1 to any real number), nor is the black one (as it does not relate 2 to any real number). As another example, > is a serial relation over the integers. But it is not a serial relation over the positive integers, because there is no y in the positive integers such that 1 > y.<a href="#cite_note-FOOTNOTEYaoWong1995-23">[19]</a> However, < is a serial relation over the positive integers, the rational numbers and the real numbers. Every reflexive relation is serial: for a given x, choose y = x.</dd></dl> <dl><dt>Surjective<a href="#cite_note-heterogeneous-17">[d]</a> (also called right-total<a href="#cite_note-kkm-18">[14]</a> or onto)</dt> <dd>For all y ∈ Y, there exists an x ∈ X such that xRy. For example, the green and blue relations in the diagram are surjective, but the red one is not (as it does not relate any real number to −1), nor is the black one (as it does not relate any real number to 2).</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Combinations_of_properties">Combinations of properties</h3>[<a href="/w/index.php?title=Relation_(mathematics)&action=edit&section=4" title="Edit section: Combinations of properties">edit</a>]</div> <dl><dd><table class="wikitable mw-collapsible" style="text-align:center;float:right;"> <caption align="top">Relations by property </caption> <tbody><tr> <th> </th> <th class="nowrap ts-vertical-header" style=""><div style=""><style data-mw-deduplicate="TemplateStyles:r1221560606">@supports(writing-mode:vertical-rl){.mw-parser-output .ts-vertical-header{line-height:1;max-width:1em;padding:0.4em;vertical-align:bottom;width:1em}html.client-js .mw-parser-output .sortable:not(.jquery-tablesorter) .ts-vertical-header:not(.unsortable),html.client-js .mw-parser-output .ts-vertical-header.headerSort{background-position:50%.4em;padding-right:0.4em;padding-top:21px}.mw-parser-output .ts-vertical-header.is-valign-top{vertical-align:top}.mw-parser-output .ts-vertical-header.is-valign-middle{vertical-align:middle}.mw-parser-output .ts-vertical-header.is-normal{font-weight:normal}.mw-parser-output .ts-vertical-header>*{display:inline-block;transform:rotate(180deg);writing-mode:vertical-rl}@supports(writing-mode:sideways-lr){.mw-parser-output .ts-vertical-header>*{transform:none;writing-mode:sideways-lr}}}</style><a href="/wiki/Reflexive_relation" title="Reflexive relation">Reflexivity</a></div> </th> <th class="nowrap ts-vertical-header" style=""><div style=""><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1221560606"><a href="/wiki/Symmetric_relation" title="Symmetric relation">Symmetry</a></div> </th> <th class="nowrap ts-vertical-header" style=""><div style=""><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1221560606"><a href="/wiki/Transitive_relation" title="Transitive relation">Transitivity</a></div> </th> <th class="nowrap ts-vertical-header" style=""><div style=""><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1221560606"><a href="/wiki/Connected_relation" title="Connected relation">Connectedness</a></div> </th> <th class="nowrap ts-vertical-header" style=""><div style=""><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1221560606">Example</div> </th></tr> <tr> <th><a href="/wiki/Partially_ordered_set#Formal_definition" title="Partially ordered set">Partial order</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Refl </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Antisym </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td> </td> <td><a href="/wiki/Subset" title="Subset">Subset</a> </td></tr> <tr> <th><a href="/wiki/Partially_ordered_set#Correspondence_of_strict_and_non-strict_partial_order_relations" title="Partially ordered set">Strict partial order</a> </th> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Irrefl </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Asym </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td> </td> <td>Strict subset </td></tr> <tr> <th><a href="/wiki/Total_order" title="Total order">Total order</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Refl </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Antisym </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td><a href="/wiki/Alphabetical_order" title="Alphabetical order">Alphabetical order</a> </td></tr> <tr> <th><a href="/wiki/Total_order#Strict_total_order" title="Total order">Strict total order</a> </th> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Irrefl </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Asym </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td>Strict alphabetical order </td></tr> <tr> <th><a href="/wiki/Equivalence_relation" title="Equivalence relation">Equivalence relation</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Refl </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Sym </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td> </td> <td><a href="/wiki/Equality_(mathematics)" title="Equality (mathematics)">Equality</a> </td></tr></tbody></table></dd></dl> Relations that satisfy certain combinations of the above properties are particularly useful, and thus have received names by their own. <dl><dt><a href="/wiki/Equivalence_relation" title="Equivalence relation">Equivalence relation</a></dt> <dd>A relation that is reflexive, symmetric, and transitive. It is also a relation that is symmetric, transitive, and serial, since these properties imply reflexivity.</dd></dl> Orderings: <dl><dt><a href="/wiki/Partially_ordered_set#Formal_definition" title="Partially ordered set">Partial order</a></dt> <dd>A relation that is reflexive, antisymmetric, and transitive.</dd></dl> <dl><dt><a href="/wiki/Partially_ordered_set#Correspondence_of_strict_and_non-strict_partial_order_relations" title="Partially ordered set">Strict partial order</a></dt> <dd>A relation that is irreflexive, asymmetric, and transitive.</dd></dl> <dl><dt><a href="/wiki/Total_order" title="Total order">Total order</a></dt> <dd>A relation that is reflexive, antisymmetric, transitive and connected.<a href="#cite_note-FOOTNOTERosenstein19824-24">[20]</a></dd></dl> <dl><dt><a href="/wiki/Total_order#Strict_total_order" title="Total order">Strict total order</a></dt> <dd>A relation that is irreflexive, asymmetric, transitive and connected.</dd></dl> Uniqueness properties: <dl><dt>One-to-one<a href="#cite_note-heterogeneous-17">[d]</a></dt> <dd>Injective and functional. For example, the green relation in the diagram is one-to-one, but the red, blue and black ones are not.</dd> <dt>One-to-many<a href="#cite_note-heterogeneous-17">[d]</a></dt> <dd>Injective and not functional. For example, the blue relation in the diagram is one-to-many, but the red, green and black ones are not.</dd> <dt>Many-to-one<a href="#cite_note-heterogeneous-17">[d]</a></dt> <dd>Functional and not injective. For example, the red relation in the diagram is many-to-one, but the green, blue and black ones are not.</dd> <dt>Many-to-many<a href="#cite_note-heterogeneous-17">[d]</a></dt> <dd>Not injective nor functional. For example, the black relation in the diagram is many-to-many, but the red, green and blue ones are not.</dd></dl> Uniqueness and totality properties: <dl><dt>A <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a><a href="#cite_note-heterogeneous-17">[d]</a></dt> <dd>A relation that is functional and total. For example, the red and green relations in the diagram are functions, but the blue and black ones are not.</dd> <dt>An <a href="/wiki/Injective_function" title="Injective function">injection</a><a href="#cite_note-heterogeneous-17">[d]</a></dt> <dd>A function that is injective. For example, the green relation in the diagram is an injection, but the red, blue and black ones are not.</dd> <dt>A <a href="/wiki/Surjective_function" title="Surjective function">surjection</a><a href="#cite_note-heterogeneous-17">[d]</a></dt> <dd>A function that is surjective. For example, the green relation in the diagram is a surjection, but the red, blue and black ones are not.</dd> <dt>A <a href="/wiki/Bijection" title="Bijection">bijection</a><a href="#cite_note-heterogeneous-17">[d]</a></dt> <dd>A function that is injective and surjective. For example, the green relation in the diagram is a bijection, but the red, blue and black ones are not.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Operations_on_relations">Operations on relations</h2>[<a href="/w/index.php?title=Relation_(mathematics)&action=edit&section=5" title="Edit section: Operations on relations">edit</a>]</div> <dl><dt>Union<a href="#cite_note-heterogeneous_operation-25">[e]</a></dt> <dd>If R and S are relations over X then R ∪ S = { (x, y) | xRy or xSy } is the union relation of R and S. The identity element of this operation is the empty relation. For example, ≤ is the union of < and =, and ≥ is the union of > and =.</dd></dl> <dl><dt> Intersection<a href="#cite_note-heterogeneous_operation-25">[e]</a></dt> <dd>If R and S are relations over X then R ∩ S = { (x, y) | xRy and xSy } is the intersection relation of R and S. The identity element of this operation is the universal relation. For example, "is a lower card of the same suit as" is the intersection of "is a lower card than" and "belongs to the same suit as".</dd></dl> <dl><dt> Composition<a href="#cite_note-heterogeneous_operation-25">[e]</a></dt> <dd>If R and S are relations over X then S ∘ R = { (x, z) | there exists y ∈ X such that xRy and ySz } (also denoted by R; S) is the <a href="/wiki/Relative_product" class="mw-redirect" title="Relative product">relative product</a> of R and S. The identity element is the identity relation. The order of R and S in the notation S ∘ R, used here agrees with the standard notational order for <a href="/wiki/Composition_of_functions" class="mw-redirect" title="Composition of functions">composition of functions</a>. For example, the composition "is mother of" ∘ "is parent of" yields "is maternal grandparent of", while the composition "is parent of" ∘ "is mother of" yields "is grandmother of". For the former case, if x is the parent of y and y is the mother of z, then x is the maternal grandparent of z.</dd></dl> <dl><dt> Converse<a href="#cite_note-heterogeneous_operation-25">[e]</a></dt> <dd>If R is a relation over sets X and Y then RT = { (y, x) | xRy } is the converse relation of R over Y and X. For example, = is the converse of itself, as is ≠, and < and > are each other's converse, as are ≤ and ≥.</dd></dl> <dl><dt> Complement<a href="#cite_note-heterogeneous_operation-25">[e]</a></dt> <dd>If R is a relation over X then R = { (x, y) | x, y ∈ X and not xRy } (also denoted by <s>R</s> or ¬R) is the complementary relation of R. For example, = and ≠ are each other's complement, as are ⊆ and ⊈, ⊇ and ⊉, and ∈ and ∉, and, for <a href="/wiki/Total_order" title="Total order">total orders</a>, also < and ≥, and > and ≤. The complement of the <a href="/wiki/Converse_relation" title="Converse relation">converse relation</a> RT is the converse of the complement: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {R^{\mathsf {T}}}}={\bar {R}}^{\mathsf {T}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>R</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {R^{\mathsf {T}}}}={\bar {R}}^{\mathsf {T}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1161afbd80cdae2d70f4533340bd2313e3f430e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.091ex; height:3.509ex;" alt="{\displaystyle {\overline {R^{\mathsf {T}}}}={\bar {R}}^{\mathsf {T}}.}"></dd></dl> <dl><dt> Restriction<a href="#cite_note-heterogeneous_operation-25">[e]</a></dt> <dd>If R is a relation over X and S is a subset of X then R|S = { (x, y) | xRy and x, y ∈ S } is the <style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style>restriction relation of R to S. The expression R|S = { (x, y) | xRy and x ∈ S } is the <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">left-restriction relation of R to S; the expression R|S = { (x, y) | xRy and y ∈ S } is called the <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">right-restriction relation of R to S. If a relation is <a href="/wiki/Reflexive_relation" title="Reflexive relation">reflexive</a>, irreflexive, <a href="/wiki/Symmetric_relation" title="Symmetric relation">symmetric</a>, <a href="/wiki/Antisymmetric_relation" title="Antisymmetric relation">antisymmetric</a>, <a href="/wiki/Asymmetric_relation" title="Asymmetric relation">asymmetric</a>, <a href="/wiki/Transitive_relation" title="Transitive relation">transitive</a>, <a href="/wiki/Serial_relation" title="Serial relation">total</a>, <a href="/wiki/Trichotomy_(mathematics)" class="mw-redirect" title="Trichotomy (mathematics)">trichotomous</a>, a <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">partial order</a>, <a href="/wiki/Total_order" title="Total order">total order</a>, <a href="/wiki/Strict_weak_order" class="mw-redirect" title="Strict weak order">strict weak order</a>, <a href="/wiki/Strict_weak_order#Total_preorders" class="mw-redirect" title="Strict weak order">total preorder</a> (weak order), or an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a>, then so too are its restrictions. However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in general not equal. For example, restricting the relation "x is parent of y" to females yields the relation "x is mother of the woman y"; its transitive closure does not relate a woman with her paternal grandmother. On the other hand, the transitive closure of "is parent of" is "is ancestor of"; its restriction to females does relate a woman with her paternal grandmother.</dd></dl> A relation R over sets X and Y is said to be <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">contained in a relation S over X and Y, written R ⊆ S, if R is a subset of S, that is, for all x ∈ X and y ∈ Y, if xRy, then xSy. If R is contained in S and S is contained in R, then R and S are called equal written R = S. If R is contained in S but S is not contained in R, then R is said to be <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">smaller than S, written R ⊊ S. For example, on the <a href="/wiki/Rational_number" title="Rational number">rational numbers</a>, the relation > is smaller than ≥, and equal to the composition > ∘ >. <div class="mw-heading mw-heading2"><h2 id="Theorems_about_relations">Theorems about relations</h2>[<a href="/w/index.php?title=Relation_(mathematics)&action=edit&section=6" title="Edit section: Theorems about relations">edit</a>]</div> <ul><li>A relation is asymmetric if, and only if, it is antisymmetric and irreflexive.</li> <li>A transitive relation is irreflexive if, and only if, it is asymmetric.</li> <li>A relation is reflexive if, and only if, its complement is irreflexive.</li> <li>A relation is strongly connected if, and only if, it is connected and reflexive.</li> <li>A relation is equal to its converse if, and only if, it is symmetric.</li> <li>A relation is connected if, and only if, its complement is anti-symmetric.</li> <li>A relation is strongly connected if, and only if, its complement is asymmetric.<a href="#cite_note-FOOTNOTESchmidtStröhlein1993-26">[21]</a></li> <li>If relation R is contained in relation S, then <ul><li>If R is reflexive, connected, strongly connected, left-total, or right-total, then so is S.</li> <li>If S is irreflexive, asymmetric, anti-symmetric, left-unique, or right-unique, then so is R.</li></ul></li> <li>A relation is reflexive, irreflexive, symmetric, asymmetric, anti-symmetric, connected, strongly connected, and transitive if its converse is, respectively.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2>[<a href="/w/index.php?title=Relation_(mathematics)&action=edit&section=7" title="Edit section: Examples">edit</a>]</div> <ul><li><a href="/wiki/Order_relation" class="mw-redirect" title="Order relation">Order relations</a>, including <a href="/wiki/Strict_order" class="mw-redirect" title="Strict order">strict orders</a>: <ul><li><a href="/wiki/Greater_than" class="mw-redirect" title="Greater than">Greater than</a></li> <li>Greater than or equal to</li> <li><a href="/wiki/Less_than" class="mw-redirect" title="Less than">Less than</a></li> <li>Less than or equal to</li> <li><a href="/wiki/Divides" class="mw-redirect" title="Divides">Divides</a> (evenly)</li> <li><a href="/wiki/Subset" title="Subset">Subset</a> of</li></ul></li> <li><a href="/wiki/Equivalence_relation" title="Equivalence relation">Equivalence relations</a>: <ul><li><a href="/wiki/Equality_(mathematics)" title="Equality (mathematics)">Equality</a></li> <li><a href="/wiki/Parallel_(geometry)" title="Parallel (geometry)">Parallel</a> with (for <a href="/wiki/Affine_space" title="Affine space">affine spaces</a>)</li> <li>Is in <a href="/wiki/Bijection" title="Bijection">bijection</a> with</li> <li><a href="/wiki/Isomorphism" title="Isomorphism">Isomorphic</a></li></ul></li> <li><a href="/wiki/Tolerance_relation" title="Tolerance relation">Tolerance relation</a>, a reflexive and symmetric relation: <ul><li><a href="/wiki/Dependency_relation" title="Dependency relation">Dependency relation</a>, a finite tolerance relation</li> <li><a href="/wiki/Independency_relation" class="mw-redirect" title="Independency relation">Independency relation</a>, the complement of some dependency relation</li></ul></li> <li><a href="/wiki/Kinship#Composition_of_relations" title="Kinship">Kinship relations</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2>[<a href="/w/index.php?title=Relation_(mathematics)&action=edit&section=8" title="Edit section: Generalizations">edit</a>]</div> The above concept of relation has been generalized to admit relations between members of two different sets. Given sets X and Y, a <a href="/wiki/Heterogeneous_relation" class="mw-redirect" title="Heterogeneous relation">heterogeneous relation</a> R over X and Y is a subset of { (x,y) | x∈X, y∈Y }.<a href="#cite_note-FOOTNOTECodd1970-2">[2]</a><a href="#cite_note-27">[22]</a> When X = Y, the relation concept described above is obtained; it is often called homogeneous relation (or endorelation)<a href="#cite_note-FOOTNOTEMüller201222-28">[23]</a><a href="#cite_note-FOOTNOTEPahlDamrath2001496-29">[24]</a> to distinguish it from its generalization. The above properties and operations that are marked "<a href="#cite_note-heterogeneous-17">[d]</a>" and "<a href="#cite_note-heterogeneous_operation-25">[e]</a>", respectively, generalize to heterogeneous relations. An example of a heterogeneous relation is "ocean x borders continent y". The best-known examples are <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a><a href="#cite_note-30">[f]</a> with distinct domains and ranges, such as sqrt : N → R+. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Relation_(mathematics)&action=edit&section=9" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Incidence_structure" title="Incidence structure">Incidence structure</a>, a heterogeneous relation between set of points and lines</li> <li><a href="/wiki/Order_theory" title="Order theory">Order theory</a>, investigates properties of order relations</li> <li><a href="/wiki/Relation_algebra" title="Relation algebra">Relation algebra</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Relation_(mathematics)&action=edit&section=10" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-10"><a href="#cite_ref-10">^</a> called "homogeneous binary relation (on sets)" when delineation from its generalizations is important </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> a generalization of sets </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> see <a href="#Properties_of_relations">below</a> </li> <li id="cite_note-heterogeneous-17">^ <a href="#cite_ref-heterogeneous_17-0">a</a> <a href="#cite_ref-heterogeneous_17-1">b</a> <a href="#cite_ref-heterogeneous_17-2">c</a> <a href="#cite_ref-heterogeneous_17-3">d</a> <a href="#cite_ref-heterogeneous_17-4">e</a> <a href="#cite_ref-heterogeneous_17-5">f</a> <a href="#cite_ref-heterogeneous_17-6">g</a> <a href="#cite_ref-heterogeneous_17-7">h</a> <a href="#cite_ref-heterogeneous_17-8">i</a> <a href="#cite_ref-heterogeneous_17-9">j</a> <a href="#cite_ref-heterogeneous_17-10">k</a> <a href="#cite_ref-heterogeneous_17-11">l</a> <a href="#cite_ref-heterogeneous_17-12">m</a> These properties also generalize to heterogeneous relations. </li> <li id="cite_note-heterogeneous_operation-25">^ <a href="#cite_ref-heterogeneous_operation_25-0">a</a> <a href="#cite_ref-heterogeneous_operation_25-1">b</a> <a href="#cite_ref-heterogeneous_operation_25-2">c</a> <a href="#cite_ref-heterogeneous_operation_25-3">d</a> <a href="#cite_ref-heterogeneous_operation_25-4">e</a> <a href="#cite_ref-heterogeneous_operation_25-5">f</a> <a href="#cite_ref-heterogeneous_operation_25-6">g</a> This operation also generalizes to heterogeneous relations. </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> that is, right-unique and left-total heterogeneous relations </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Relation_(mathematics)&action=edit&section=11" title="Edit section: References">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</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="CITEREFStoll" class="citation book cs1">Stoll, Robert R. 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San Francisco, CA: Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-63829-4" title="Special:BookSources/978-0-486-63829-4"><bdi>978-0-486-63829-4</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=San+Francisco%2C+CA&rft.pub=Dover+Publications&rft.isbn=978-0-486-63829-4&rft.aulast=Stoll&rft.aufirst=Robert+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARelation+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-FOOTNOTECodd1970-2">^ <a href="#cite_ref-FOOTNOTECodd1970_2-0">a</a> <a href="#cite_ref-FOOTNOTECodd1970_2-1">b</a> <a href="#cite_ref-FOOTNOTECodd1970_2-2">c</a> <a href="#CITEREFCodd1970">Codd 1970</a> </li> <li id="cite_note-FOOTNOTEHalmos1968Ch_14-3"><a href="#cite_ref-FOOTNOTEHalmos1968Ch_14_3-0">^</a> <a href="#CITEREFHalmos1968">Halmos 1968</a>, Ch 14 </li> <li id="cite_note-FOOTNOTEHalmos1968Ch_7-4"><a href="#cite_ref-FOOTNOTEHalmos1968Ch_7_4-0">^</a> <a href="#CITEREFHalmos1968">Halmos 1968</a>, Ch 7 </li> <li id="cite_note-5"><a href="#cite_ref-5">^</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://mathinsight.org/definition/relation">"Relation definition – Math Insight"</a>. mathinsight.org. 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I. Lewis">C. I. Lewis</a> (1918) <a rel="nofollow" class="external text" href="https://archive.org/details/asurveyofsymboli00lewiuoft">A Survey of Symbolic Logic</a>, pp. 269–279, via internet Archive </li> <li id="cite_note-FOOTNOTESchmidt2010Chapt._5-9">^ <a href="#cite_ref-FOOTNOTESchmidt2010Chapt._5_9-0">a</a> <a href="#cite_ref-FOOTNOTESchmidt2010Chapt._5_9-1">b</a> <a href="#CITEREFSchmidt2010">Schmidt 2010</a>, Chapt. 5 </li> <li id="cite_note-FOOTNOTEEnderton1977Ch_3._p._40-12"><a href="#cite_ref-FOOTNOTEEnderton1977Ch_3._p._40_12-0">^</a> <a href="#CITEREFEnderton1977">Enderton 1977</a>, Ch 3. p. 40 </li> <li id="cite_note-FOOTNOTESmithEggenSt._Andre2006160-14"><a href="#cite_ref-FOOTNOTESmithEggenSt._Andre2006160_14-0">^</a> <a href="#CITEREFSmithEggenSt._Andre2006">Smith, Eggen & St. Andre 2006</a>, p. 160 </li> <li id="cite_note-FOOTNOTENievergelt2002[httpsbooksgooglecombooksid_H_nJdagqL8CpgPA158_158]-15"><a href="#cite_ref-FOOTNOTENievergelt2002[httpsbooksgooglecombooksid_H_nJdagqL8CpgPA158_158]_15-0">^</a> <a href="#CITEREFNievergelt2002">Nievergelt 2002</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_H_nJdagqL8C&pg=PA158">158</a> </li> <li id="cite_note-FOOTNOTEFlaškaJežekKepkaKortelainen2007p.1_Lemma_1.1_(iv)._This_source_refers_to_asymmetric_relations_as_"strictly_antisymmetric".-16"><a href="#cite_ref-FOOTNOTEFlaškaJežekKepkaKortelainen2007p.1_Lemma_1.1_(iv)._This_source_refers_to_asymmetric_relations_as_"strictly_antisymmetric"._16-0">^</a> <a href="#CITEREFFlaškaJežekKepkaKortelainen2007">Flaška et al. 2007</a>, p.1 Lemma 1.1 (iv). This source refers to asymmetric relations as "strictly antisymmetric". </li> <li id="cite_note-kkm-18">^ <a href="#cite_ref-kkm_18-0">a</a> <a href="#cite_ref-kkm_18-1">b</a> <a href="#cite_ref-kkm_18-2">c</a> <a href="#CITEREFKilpKnauerMikhalev2000">Kilp, Knauer & Mikhalev 2000</a>, p. 3. The same four definitions appear in the following: <a href="#CITEREFPahlDamrath2001">Pahl & Damrath 2001</a>, p. 506, <a href="#CITEREFBest1996">Best 1996</a>, pp. 19–21, <a href="#CITEREFRiemann1999">Riemann 1999</a>, pp. 21–22 </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> Van Gasteren 1990, p. 45. </li> <li id="cite_note-20"><a href="#cite_ref-20">^</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://encyclopediaofmath.org/wiki/Functional_relation">"Functional relation - Encyclopedia of Mathematics"</a>. encyclopediaofmath.org. Retrieved 2024-06-13.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=encyclopediaofmath.org&rft.atitle=Functional+relation+-+Encyclopedia+of+Mathematics&rft_id=https%3A%2F%2Fencyclopediaofmath.org%2Fwiki%2FFunctional_relation&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARelation+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-21"><a href="#cite_ref-21">^</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://ncatlab.org/nlab/show/functional+relation">"functional relation in nLab"</a>. ncatlab.org. Retrieved 2024-06-13.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=ncatlab.org&rft.atitle=functional+relation+in+nLab&rft_id=https%3A%2F%2Fncatlab.org%2Fnlab%2Fshow%2Ffunctional%2Brelation&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARelation+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-FOOTNOTEMäs2007-22"><a href="#cite_ref-FOOTNOTEMäs2007_22-0">^</a> <a href="#CITEREFMäs2007">Mäs 2007</a> </li> <li id="cite_note-FOOTNOTEYaoWong1995-23"><a href="#cite_ref-FOOTNOTEYaoWong1995_23-0">^</a> <a href="#CITEREFYaoWong1995">Yao & Wong 1995</a> </li> <li id="cite_note-FOOTNOTERosenstein19824-24"><a href="#cite_ref-FOOTNOTERosenstein19824_24-0">^</a> <a href="#CITEREFRosenstein1982">Rosenstein 1982</a>, p. 4 </li> <li id="cite_note-FOOTNOTESchmidtStröhlein1993-26"><a href="#cite_ref-FOOTNOTESchmidtStröhlein1993_26-0">^</a> <a href="#CITEREFSchmidtStröhlein1993">Schmidt & Ströhlein 1993</a> </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> <a href="#CITEREFEnderton1977">Enderton 1977</a>, Ch 3. p. 40 </li> <li id="cite_note-FOOTNOTEMüller201222-28"><a href="#cite_ref-FOOTNOTEMüller201222_28-0">^</a> <a href="#CITEREFMüller2012">Müller 2012</a>, p. 22 </li> <li id="cite_note-FOOTNOTEPahlDamrath2001496-29"><a href="#cite_ref-FOOTNOTEPahlDamrath2001496_29-0">^</a> <a href="#CITEREFPahlDamrath2001">Pahl & Damrath 2001</a>, p. 496 </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2>[<a href="/w/index.php?title=Relation_(mathematics)&action=edit&section=12" title="Edit section: Bibliography">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBest1996" class="citation book cs1"><a href="/wiki/Eike_Best" title="Eike Best">Best, Eike</a> (1996). 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Communications of the ACM. 13 (6): 377–387. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F362384.362685">10.1145/362384.362685</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:207549016">207549016</a>. Retrieved 2020-04-29.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+of+the+ACM&rft.atitle=A+Relational+Model+of+Data+for+Large+Shared+Data+Banks&rft.volume=13&rft.issue=6&rft.pages=377-387&rft.date=1970-06&rft_id=info%3Adoi%2F10.1145%2F362384.362685&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A207549016%23id-name%3DS2CID&rft.aulast=Codd&rft.aufirst=Edgar+Frank&rft_id=https%3A%2F%2Fwww.seas.upenn.edu%2F~zives%2F03f%2Fcis550%2Fcodd.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARelation+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCodd1990" class="citation book cs1"><a href="/wiki/Edgar_F._Codd" title="Edgar F. Codd">Codd, Edgar Frank</a> (1990). <a rel="nofollow" class="external text" href="https://dl.acm.org/doi/pdf/10.5555/77708">The Relational Model for Database Management: Version 2</a>. Boston: <a href="/wiki/Addison-Wesley" title="Addison-Wesley">Addison-Wesley</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0201141924" title="Special:BookSources/978-0201141924"><bdi>978-0201141924</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+Relational+Model+for+Database+Management%3A+Version+2&rft.place=Boston&rft.pub=Addison-Wesley&rft.date=1990&rft.isbn=978-0201141924&rft.aulast=Codd&rft.aufirst=Edgar+Frank&rft_id=https%3A%2F%2Fdl.acm.org%2Fdoi%2Fpdf%2F10.5555%2F77708&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARelation+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEnderton1977" class="citation book cs1"><a href="/wiki/Herbert_Enderton" title="Herbert Enderton">Enderton, Herbert</a> (1977). Elements of Set Theory. Boston: <a href="/wiki/Academic_Press" title="Academic Press">Academic Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-238440-0" title="Special:BookSources/978-0-12-238440-0"><bdi>978-0-12-238440-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+Set+Theory&rft.place=Boston&rft.pub=Academic+Press&rft.date=1977&rft.isbn=978-0-12-238440-0&rft.aulast=Enderton&rft.aufirst=Herbert&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARelation+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFlaškaJežekKepkaKortelainen2007" class="citation book cs1">Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. (2007). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20131102214049/http://www.karlin.mff.cuni.cz/~jezek/120/transitive1.pdf">Transitive Closures of Binary Relations I</a> (PDF). Prague: School of Mathematics – Physics Charles University. 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Princeton: Nostrand.</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&rft.pub=Nostrand&rft.date=1968&rft.aulast=Halmos&rft.aufirst=Paul+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARelation+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKilpKnauerMikhalev2000" class="citation book cs1">Kilp, Mati; Knauer, Ulrich; Mikhalev, Alexander (2000). Monoids, Acts and Categories: with Applications to Wreath Products and Graphs. Berlin: <a href="/wiki/Walter_de_Gruyter" class="mw-redirect" title="Walter de Gruyter">De Gruyter</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-11-015248-7" title="Special:BookSources/978-3-11-015248-7"><bdi>978-3-11-015248-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Monoids%2C+Acts+and+Categories%3A+with+Applications+to+Wreath+Products+and+Graphs&rft.place=Berlin&rft.pub=De+Gruyter&rft.date=2000&rft.isbn=978-3-11-015248-7&rft.aulast=Kilp&rft.aufirst=Mati&rft.au=Knauer%2C+Ulrich&rft.au=Mikhalev%2C+Alexander&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARelation+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMäs2007" class="citation cs2">Mäs, Stephan (2007), "Reasoning on Spatial Semantic Integrity Constraints", Spatial Information Theory: 8th International Conference, COSIT 2007, Melbourne, Australia, September 19–23, Proceedings, Lecture Notes in Computer Science, vol. 4736, Springer, pp. 285–302, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-540-74788-8_18">10.1007/978-3-540-74788-8_18</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Reasoning+on+Spatial+Semantic+Integrity+Constraints&rft.btitle=Spatial+Information+Theory%3A+8th+International+Conference%2C+COSIT+2007%2C+Melbourne%2C+Australia%2C+September+19%E2%80%9323%2C+Proceedings&rft.series=Lecture+Notes+in+Computer+Science&rft.pages=285-302&rft.pub=Springer&rft.date=2007&rft_id=info%3Adoi%2F10.1007%2F978-3-540-74788-8_18&rft.aulast=M%C3%A4s&rft.aufirst=Stephan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARelation+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMüller2012" class="citation book cs1">Müller, M. 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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/978-3-540-67995-0" title="Special:BookSources/978-3-540-67995-0"><bdi>978-3-540-67995-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Foundations+of+Computational+Engineering%3A+A+Handbook&rft.pub=Springer+Science+%26+Business+Media&rft.date=2001&rft.isbn=978-3-540-67995-0&rft.aulast=Pahl&rft.aufirst=Peter+J.&rft.au=Damrath%2C+Rudolf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARelation+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeirce1873" class="citation journal cs1"><a href="/wiki/Charles_Sanders_Peirce" title="Charles Sanders Peirce">Peirce, Charles Sanders</a> (1873). <a rel="nofollow" class="external text" href="https://archive.org/details/descriptionanot00peirgoog/mode/2up">"Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic"</a>. Memoirs of the American Academy of Arts and Sciences. 9 (2): 317–178. <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/1873MAAAS...9..317P">1873MAAAS...9..317P</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F25058006">10.2307/25058006</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/2027%2Fhvd.32044019561034">2027/hvd.32044019561034</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/25058006">25058006</a>. Retrieved 2020-05-05.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Memoirs+of+the+American+Academy+of+Arts+and+Sciences&rft.atitle=Description+of+a+Notation+for+the+Logic+of+Relatives%2C+Resulting+from+an+Amplification+of+the+Conceptions+of+Boole%27s+Calculus+of+Logic&rft.volume=9&rft.issue=2&rft.pages=317-178&rft.date=1873&rft_id=info%3Ahdl%2F2027%2Fhvd.32044019561034&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F25058006%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F25058006&rft_id=info%3Abibcode%2F1873MAAAS...9..317P&rft.aulast=Peirce&rft.aufirst=Charles+Sanders&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdescriptionanot00peirgoog%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARelation+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRiemann1999" class="citation book cs1">Riemann, Robert-Christoph (1999). Modelling of Concurrent Systems: Structural and Semantical Methods in the High Level Petri Net Calculus. Herbert Utz Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-89675-629-9" title="Special:BookSources/978-3-89675-629-9"><bdi>978-3-89675-629-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modelling+of+Concurrent+Systems%3A+Structural+and+Semantical+Methods+in+the+High+Level+Petri+Net+Calculus&rft.pub=Herbert+Utz+Verlag&rft.date=1999&rft.isbn=978-3-89675-629-9&rft.aulast=Riemann&rft.aufirst=Robert-Christoph&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARelation+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRosenstein1982" class="citation cs2">Rosenstein, Joseph G. (1982), Linear orderings, Academic Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-597680-1" title="Special:BookSources/0-12-597680-1"><bdi>0-12-597680-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+orderings&rft.pub=Academic+Press&rft.date=1982&rft.isbn=0-12-597680-1&rft.aulast=Rosenstein&rft.aufirst=Joseph+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARelation+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchmidt2010" class="citation book cs1"><a href="/wiki/Gunther_Schmidt" title="Gunther Schmidt">Schmidt, Gunther</a> (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=E4dREBTs5WsC">Relational Mathematics</a>. Cambridge: <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/978-0-521-76268-7" title="Special:BookSources/978-0-521-76268-7"><bdi>978-0-521-76268-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Relational+Mathematics&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=2010&rft.isbn=978-0-521-76268-7&rft.aulast=Schmidt&rft.aufirst=Gunther&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DE4dREBTs5WsC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARelation+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchmidtStröhlein1993" class="citation book cs1"><a href="/wiki/Gunther_Schmidt" title="Gunther Schmidt">Schmidt, Gunther</a>; Ströhlein, Thomas (1993). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZgarCAAAQBAJ">Relations and Graphs: Discrete Mathematics for Computer Scientists</a>. Berlin: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-77970-1" title="Special:BookSources/978-3-642-77970-1"><bdi>978-3-642-77970-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Relations+and+Graphs%3A+Discrete+Mathematics+for+Computer+Scientists&rft.place=Berlin&rft.pub=Springer&rft.date=1993&rft.isbn=978-3-642-77970-1&rft.aulast=Schmidt&rft.aufirst=Gunther&rft.au=Str%C3%B6hlein%2C+Thomas&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DZgarCAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARelation+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmithEggenSt._Andre2006" class="citation cs2">Smith, Douglas; Eggen, Maurice; St. Andre, Richard (2006), A Transition to Advanced Mathematics (6th ed.), Brooks/Cole, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-534-39900-2" title="Special:BookSources/0-534-39900-2"><bdi>0-534-39900-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Transition+to+Advanced+Mathematics&rft.edition=6th&rft.pub=Brooks%2FCole&rft.date=2006&rft.isbn=0-534-39900-2&rft.aulast=Smith&rft.aufirst=Douglas&rft.au=Eggen%2C+Maurice&rft.au=St.+Andre%2C+Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARelation+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVan_Gasteren1990" class="citation book cs1">Van Gasteren, Antonetta (1990). On the Shape of Mathematical Arguments. Berlin: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783540528494" title="Special:BookSources/9783540528494"><bdi>9783540528494</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=On+the+Shape+of+Mathematical+Arguments&rft.place=Berlin&rft.pub=Springer&rft.date=1990&rft.isbn=9783540528494&rft.aulast=Van+Gasteren&rft.aufirst=Antonetta&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARelation+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYaoWong1995" class="citation journal cs1">Yao, Y.Y.; Wong, S.K.M. (1995). <a rel="nofollow" class="external text" href="http://www2.cs.uregina.ca/~yyao/PAPERS/relation.pdf">"Generalization of rough sets using relationships between attribute values"</a> (PDF). Proceedings of the 2nd Annual Joint Conference on Information Sciences: 30–33.</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+2nd+Annual+Joint+Conference+on+Information+Sciences&rft.atitle=Generalization+of+rough+sets+using+relationships+between+attribute+values&rft.pages=30-33&rft.date=1995&rft.aulast=Yao&rft.aufirst=Y.Y.&rft.au=Wong%2C+S.K.M.&rft_id=http%3A%2F%2Fwww2.cs.uregina.ca%2F~yyao%2FPAPERS%2Frelation.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARelation+%28mathematics%29" class="Z3988"></li></ul> </div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline 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href="/wiki/Template:Functions_navbox" title="Template:Functions navbox"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template_talk:Functions_navbox&action=edit&redlink=1" class="new" title="Template talk:Functions navbox (page does not exist)"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Functions_navbox" title="Special:EditPage/Template:Functions navbox"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Function" style="font-size:114%;margin:0 4em"><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Function</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/History_of_the_function_concept" title="History of the function concept">History</a></li> <li><a href="/wiki/List_of_mathematical_functions" title="List of mathematical functions">List of specific functions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types by domain and codomain</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Boolean-valued_function" title="Boolean-valued function">X → 𝔹</a></li> <li><a href="/wiki/Ordered_pair" title="Ordered pair">𝔹 → X</a></li> <li><a href="/wiki/Boolean_function" title="Boolean function">𝔹ⁿ → X</a></li> <li><a href="/wiki/Integer-valued_function" title="Integer-valued function">X → ℤ</a></li> <li><a href="/wiki/Sequence" title="Sequence">ℤ → X</a></li> <li><a href="/wiki/Real-valued_function" title="Real-valued function">X → ℝ</a></li> <li><a href="/wiki/Function_of_a_real_variable" title="Function of a real variable">ℝ → X</a></li> <li><a href="/wiki/Function_of_several_real_variables" title="Function of several real variables">ℝⁿ → X</a></li> <li><a href="/wiki/Complex-valued_function" class="mw-redirect" title="Complex-valued function">X → ℂ</a></li> <li><a href="/wiki/Function_of_a_complex_variable" class="mw-redirect" title="Function of a complex variable">ℂ → X</a></li> <li><a href="/wiki/Function_of_several_complex_variables" title="Function of several complex variables">ℂⁿ → X</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Classes/properties</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Constant_function" title="Constant function">Constant</a></li> <li><a href="/wiki/Identity_function" title="Identity function">Identity</a></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear</a></li> <li><a href="/wiki/Polynomial" title="Polynomial">Polynomial</a></li> <li><a href="/wiki/Rational_function" title="Rational function">Rational</a></li> <li><a href="/wiki/Algebraic_function" title="Algebraic function">Algebraic</a></li> <li><a href="/wiki/Analytic_function" title="Analytic function">Analytic</a></li> <li><a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">Smooth</a></li> <li><a href="/wiki/Continuous_function" title="Continuous function">Continuous</a></li> <li><a href="/wiki/Measurable_function" title="Measurable function">Measurable</a></li> <li><a href="/wiki/Injective_function" title="Injective function">Injective</a></li> <li><a href="/wiki/Surjective_function" title="Surjective function">Surjective</a></li> <li><a href="/wiki/Bijection" title="Bijection">Bijective</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constructions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Restriction_(mathematics)" title="Restriction (mathematics)">Restriction</a></li> <li><a href="/wiki/Function_composition" title="Function composition">Composition</a></li> <li><a href="/wiki/Lambda_calculus" title="Lambda calculus">λ</a></li> <li><a href="/wiki/Inverse_function" title="Inverse function">Inverse</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Relation</a> (<a href="/wiki/Binary_relation" title="Binary relation">Binary relation</a>)</li> <li><a href="/wiki/Set-valued_function" title="Set-valued function">Set-valued</a></li> <li><a href="/wiki/Multivalued_function" title="Multivalued function">Multivalued</a></li> <li><a href="/wiki/Partial_function" title="Partial function">Partial</a></li> <li><a href="/wiki/Implicit_function" title="Implicit function">Implicit</a></li> <li><a href="/wiki/Function_space" title="Function space">Space</a></li> <li><a href="/wiki/Higher-order_function" title="Higher-order function">Higher-order</a></li> <li><a href="/wiki/Morphism" title="Morphism">Morphism</a></li> <li><a href="/wiki/Functor" title="Functor">Functor</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Category:Functions" title="Category:Functions">Category</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: 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