Equivalence relations

In mathematics, an equivalence relation is a binary relation that is at the same time a reflexive relation, a symmetric relation and a transitive relation. As a consequence of these properties an equivalence relation provides a partition of a set into equivalence classes.

Although various notations are used throughout the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R, the most common are "a ~ b" and "a ≡ b", which are used when R is the obvious relation being referenced, and variations of "a ~_Rb", "a ≡_Rb", or "aRb" otherwise.

A given binary relation ~ on a set X is said to be an equivalence relation if and only if it is reflexive, symmetric and transitive. That is, for all a, b and c in X:

X together with the relation ~ is called a setoid. The equivalence class of ${\displaystyle a}$ under ~, denoted ${\displaystyle [a]}$, is defined as ${\displaystyle [a]=\{b\in X\mid a\sim b\}}$.

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