Equivalence relation

An equivalence relation is a binary relation on a set that groups elements into categories in which all members are considered "equivalent" under some criterion.

Definition

A relation $\sim$ on set $X$ is a equivalence relation if it satisfies the following properties:

• Reflexivity: $\mathrm{\forall }x\in X$, $x\sim x$.
• Symmetry: $\mathrm{\forall }x,y\in X$ such that $x\sim y$, $y\sim x$.
• Transitivity: $\mathrm{\forall }x,y,z$, if $x\sim y$ and $y\sim z$, then $x\sim z$.

When $x\sim y$, "$x$ is said to be equivalent to $y$" under the relation $\sim$.

Equivalence classes

Given $x\in X$, the equivalence class of $x$, denoted $$[x]=\{y\in X\mid y\sim x\}$$ is the set of elements that are equivalent to $x$.