Set/Equivalence relation/Definition

Equivalence relation

An equivalence relation on a set ${}M$ is a relation ${}R\subseteq M\times M$ fulfilling the following three properties (for arbitrary ${}x,y,z\in M$).

1. ${}x\sim x$ (reflexive).
2. ${}x\sim y$ implies ${}y\sim x$ (symmetric).
3. From ${}x\sim y$ and ${}y\sim z$ we can deduce ${}x\sim z$ (transitive).

Here, ${}x\sim y$ means that the pair ${}(x,y)$ belongs to ${}R$.