Equivalence class

Mathematical concept

In mathematics, when the elements of some set S {\displaystyle S} have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S {\displaystyle S} into equivalence classes. These equivalence classes are constructed so that elements a {\displaystyle a} and b {\displaystyle b} belong to the same equivalence class if, and only if, they are equivalent. Formally, given a set S {\displaystyle S} and an equivalence relation ∼ {\displaystyle \sim } on S , {\displaystyle S,} the equivalence class of an element a {\displaystyle a} in S {\displaystyle S} is denoted [ a ] {\displaystyle [a]} or, equivalently, [ a ] ∼ {\displaystyle [a]_{\sim }} to emphasize its equivalence relation ∼ {\displaystyle \sim } , and is defined as the set of all elements in S {\displaystyle S} with which a {\displaystyle a} is ∼ {\displaystyle \sim } -related.

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