Equivalence relation

Equivalence relation --- отношение эквивалентности.

A relation is an equivalence relation if it is reflexive, symmetric, and transitive. For an equivalence relation [math]\displaystyle{ R }[/math] in a set [math]\displaystyle{ S }[/math], the equivalence class of [math]\displaystyle{ a }[/math] with respect to [math]\displaystyle{ R }[/math] is the set of all elements [math]\displaystyle{ b }[/math] in [math]\displaystyle{ S }[/math] such that [math]\displaystyle{ aRb }[/math]. When the relation [math]\displaystyle{ R }[/math] is understood, the equivalence class of [math]\displaystyle{ a }[/math] is denoted by [math]\displaystyle{ [a] }[/math]. The equivalence classes of an equivalence relation in [math]\displaystyle{ S }[/math] form a partition of [math]\displaystyle{ S }[/math]. If an equivalence relation [math]\displaystyle{ R_{1} }[/math] is contained in another equivalence relation [math]\displaystyle{ R_{2} }[/math] (i.e., if [math]\displaystyle{ aR_{1}b }[/math] implies [math]\displaystyle{ aR_{2}b }[/math]), then the partition formed by the equivalence classes with respect to [math]\displaystyle{ R_{1} }[/math] is finer than the partition formed by the classes with respect to [math]\displaystyle{ R_{2} }[/math].