Editing Equivalence relation (section)

=== Alternative definition using relational algebra ===

In [[relational algebra]], if <math>R\subseteq X\times Y</math> and <math>S\subseteq Y\times Z</math> are relations, then the [[Composition of relations|composite relation]] <math>SR\subseteq X\times Z</math> is defined so that <math>x \, SR \, z</math> if and only if there is a <math>y\in Y</math> such that <math>x \, R \, y</math> and <math>y \, S \, z</math>.<ref group="note">Sometimes the composition <math>SR\subseteq X\times Z</math> is instead written as <math>R;S</math>, or as <math>RS</math>; in both cases, <math>R</math> is the first relation that is applied. See the article on [[Composition of relations#Notational variations|Composition of relations]] for more information.</ref> This definition is a generalisation of the definition of [[Function composition|functional composition]]. The defining properties of an equivalence relation <math>R</math> on a set <math>X</math> can then be reformulated as follows:
* <math>\operatorname{id} \subseteq R</math>. ([[Reflexive relation|reflexivity]]). (Here, <math>\operatorname{id}</math> denotes the [[identity function]] on <math>X</math>.)
* <math>R=R^{-1}</math> ([[Symmetric relation|symmetry]]).
* <math>RR\subseteq R</math> ([[Transitive relation|transitivity]]).<ref>{{Cite book |last=Halmos |first=Paul Richard |title=Naive Set Theory |publisher=Springer |year=1914 |isbn=978-0-387-90104-6 |location=New York |pages=41 |language=English}}</ref>