# Equivalence relation

2010 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.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Equivalence&oldid=30492