An equivalence relation is a relation that is reflexive, symmetric, and transitive. In other words, it is a relation that defines a partitioning of a set into equivalence classes.An example of an equivalence relation is "is equal to" on the set of real numbers. This relation is reflexive because every number is equal to itself, symmetric because if a = b then b = a, and transitive because if a = b and b = c then a = c.Another example of an equivalence relation is "has the same parity" on the set of natural numbers. This relation is reflexive because every number is of the same parity as itself, symmetric because if a and b have the same parity, then b and a do as well, and transitive because if a and b have the same parity and b and c have the same parity then a and c have the same parity.In general, given a set A, an equivalence relation on A is a relation R on A such that 1. it is reflexive, i.e., (a,a) ∈ R for all a ∈ A 2. it is symmetric, i.e., (a,b) ∈ R implies (b,a) ∈ R 3. it is transitive, i.e., (a,b) ∈ R and (b,c) ∈ R implies (a,c) ∈ REquivalence relations have many important uses in mathematics, including in number theory, group theory, and topology.