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Difference between revisions of "Equivalence relation"

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A [[Binary relation|binary relation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036030/e0360301.png" /> on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036030/e0360302.png" /> with the following properties:
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{{TEX|done}}
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{{MSC|03E}}
  
1) for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036030/e0360303.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036030/e0360304.png" /> (reflexivity);
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Let $X$ be a set. An equivalence relation on $X$ is a subset $R\subseteq X\times X$ that satisfies the following three properties:
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036030/e0360305.png" /> (symmetry);
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1) [[Reflexivity]]: for all $x\in X$, $(x,x)\in R$;
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036030/e0360306.png" /> (transitivity).
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2) [[Symmetry (of a relation)|Symmetry]]: for all $x,y\in X$, if $(x,y)\in R$ then $(y,x)\in R$;
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036030/e0360307.png" /> maps the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036030/e0360308.png" /> into a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036030/e0360309.png" />, then the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036030/e03603010.png" /> is an equivalence.
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3) [[Transitive relation|Transitivity]]: for all $x,y,z \in X$, if $(x,y)\in R$ and  $(y,z)\in R$ then $(x,z)\in R$.
  
For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036030/e03603011.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036030/e03603012.png" /> consisting of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036030/e03603013.png" /> equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036030/e03603014.png" /> is called the equivalence class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036030/e03603015.png" />. Any two equivalence classes either are disjoint or coincide, that is, any equivalence defines a partition (decomposition) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036030/e03603016.png" />, and vice versa.
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When $(x,y)\in R$ we say that $x$ is equivalent to $y$.
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Instead of $(x,y)\in R$, the notation $xRy$, or even $x\sim y$,  is also used.  
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An equivalence relation is a [[Binary relation|binary relation]].  
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Example: If $f$ maps the set $X$ into a set $Y$, then $R=\{(x_1,x_2)\in X\times X\,:\, f(x_1)=f(x_2)\}$ is an equivalence relation (cf. [[Kernel of a function]]).
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For any $y\in X$ the subset  of $X$ that  consists of all $x$ that are equivalent to $y$ is called the equivalence class of $y$, denoted $[y]_R$. Any two equivalence classes either are disjoint or coincide, that is, any equivalence relation on $X$ defines a [[partition]] (decomposition) of $X$, and vice versa.
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A subset $A$ of $X$ is ''saturated'' with respect to $R$ if $a \in A,\, b \in X,\,a R b \Rightarrow b \in A$: so $A$ is saturated if and only if it is a union of equivalence classes of $R$.  For any subset $A$ of $X$, the ''saturation'' $A^R$ is the set $\{ b \in X : \exists a \in A,\,a R b \} = \cup_{a \in A} [a]_R$, the union of the equivalence classes that meet $A$.
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====References====
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* N. Bourbaki, ''Theory of Sets'', Eléments de mathématiques '''1''', reprint ed. Springer (2004) {{ISBN|3-540-22525-0}} {{ZBL|1061.03001}}

Latest revision as of 18:10, 14 October 2023

2020 Mathematics Subject Classification: Primary: 03E [MSN][ZBL]

Let $X$ be a set. An equivalence relation on $X$ is a subset $R\subseteq X\times X$ that satisfies the following three properties:

1) Reflexivity: for all $x\in X$, $(x,x)\in R$;

2) Symmetry: for all $x,y\in X$, if $(x,y)\in R$ then $(y,x)\in R$;

3) Transitivity: for all $x,y,z \in X$, if $(x,y)\in R$ and $(y,z)\in R$ then $(x,z)\in R$.

When $(x,y)\in R$ we say that $x$ is equivalent to $y$.

Instead of $(x,y)\in R$, the notation $xRy$, or even $x\sim y$, is also used.

An equivalence relation is a binary relation.

Example: If $f$ maps the set $X$ into a set $Y$, then $R=\{(x_1,x_2)\in X\times X\,:\, f(x_1)=f(x_2)\}$ is an equivalence relation (cf. Kernel of a function).

For any $y\in X$ the subset of $X$ that consists of all $x$ that are equivalent to $y$ is called the equivalence class of $y$, denoted $[y]_R$. Any two equivalence classes either are disjoint or coincide, that is, any equivalence relation on $X$ defines a partition (decomposition) of $X$, and vice versa.

A subset $A$ of $X$ is saturated with respect to $R$ if $a \in A,\, b \in X,\,a R b \Rightarrow b \in A$: so $A$ is saturated if and only if it is a union of equivalence classes of $R$. For any subset $A$ of $X$, the saturation $A^R$ is the set $\{ b \in X : \exists a \in A,\,a R b \} = \cup_{a \in A} [a]_R$, the union of the equivalence classes that meet $A$.

References

  • N. Bourbaki, Theory of Sets, Eléments de mathématiques 1, reprint ed. Springer (2004) ISBN 3-540-22525-0 Zbl 1061.03001
How to Cite This Entry:
Equivalence relation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equivalence_relation&oldid=17622
This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article