equivalence class

Let $S$ be a set with an equivalence relation $\sim$ . An equivalence class of $S$ under $\sim$ is a subset $T\subset S$ such that

• If $x \in T$ and $y \in S$ , then $x \sim y$ if and only if $y \in T$
• If $S$ is nonempty, then $T$ is nonempty

For $x \in S$ , the equivalence class containing $x$ is often denoted by $[x]$ , so that $$ [x] := \{ y \in S \mid x \sim y \}. $$

The set of all equivalence classes of $S$ under $\sim$ is defined to be the set of all subsets of $S$ which are equivalence classes of $S$ under $\sim$ , and is denoted by $S/\sim$ . The map $x\mapsto [x]$ is sometimes referred to as the canonical projection.

For any equivalence relation $\sim$ , the set of all equivalence classes of $S$ under $\sim$ is a partition of $S$ , and this correspondence is a bijection between the set of equivalence relations on $S$ and the set of partitions of $S$ (consisting of nonempty sets).