Last modified on 13 July 2016, at 12:04

Equivalence relation

An equivalence relation is a mathematical relation (denoted by ~) which is reflexive, symmetric, and transitive.

1. Reflexive: a~a.

2. Symmetric: if a~b then b~a.

3. Transitive: if a~b and b~c, then a~c.

Equality and set membership are the most basic equivalence relations, which is why they are often called trivial equivalence relations. Every equivalence relation is constructed from irreducible equivalences.

Congruence relations are examples of equivalence relations, as is the j-invariant of elliptic curves.

An equivalence relation on a set S partitions S into disjoint subsets, known as equivalence classes. Two objects in the same equivalence class are said to be equivalent.

Equivalence relations are the mathematical foundation for forming quotients.