# Equivalence relation

**Equivalence relation**as edited by ReligiousRight (Talk | contribs) at 22:24, 1 June 2010. This URL is a permanent link to this version of this page.

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.