A binary relation $\sim$ on a set $X$ is an equivalence relation if:
Reflexivitiy. For all $x\in X$,
\[x \sim x.\]Symmetry. For all $x$, $y\in X$,
\[x \sim y \enspace\rimp\enspace y \sim x.\]Transitivity. For all $x$, $y$, $z\in X$,
\[x \sim y \,\land\, y \sim z \enspace\rimp\enspace x \sim z.\]
Let $\sim$ be an equivalence relation on a set $X$ and $a \in X$. The equivalence class of $a$ with respect to $\sim$ is
\[[a]_\sim = \{x \in X : a \sim x\}.\]If there is no ambiguity, $[a]_\sim$ can be simply written as $[a]$.
Let $\sim$ be an equivalence relation on a set $X$. The quotient set of $X$ by $\sim$ is the set
\[X / {\sim} \enspace=\enspace \{[x] : x \in X\}.\]
Let $\sim$ be an equivalence relation on a set $X$. For any $x$, $y \in X$, the following statements are equivalent:
- \[x \sim y.\]
- \[[x] = [y].\]
- \[x \in [y].\]
- \[[x] \cap[y] \neq \empt.\]