Equivalence relation

An equivalence relation \((\sim)\) is a relation on a set that is reflexive, symmetric, and transitive. This type of relation partitions its underlying set into disjoint equivalence classes.

A relation \(\sim\) is an equivalence relation if and only if the following holds for all \(a, b, c \in X\):
  • Reflexive: \(a \sim a\)
  • Symmetric: \(a \sim b\) if and only if \(b \sim a\)
  • Transitive: If \(a \sim b\) and \(b \sim c\), then \(a \sim c\)

Equality \((=)\) is perhaps the most well-known equivalence relation. Congruence modulo \(n\) \((\equiv_{n})\) is also an equivalence relation. Rhyming is an example of an equivalence relation on a set of words.

To disprove that a relation is an equivalence relation, you only need to come up with a single counterexample to one of its three defining properties.

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