Equivalence relation

Definition · applies to relations

Undergraduate

Consider a set $X$ and a relation $\sim$ on $X$. The relation $\sim$ is an equivalence relation if, for all $a, b, c \in X$,

1. $a \sim a$ (reflexive),
2. $a \sim b$ implies $b \sim a$ (symmetric), and
3. $a \sim b$ and $b \sim c$ imply $a \sim c$ (transitive).

Properties

• An equivalence relation splits $X$ into disjoint equivalence classes $[a] = \setdef{b \in X \suchthat b \sim a}$, which form a partition of $X$, and every partition comes from an equivalence relation in this way.

All definitions · All distinctions · All examples