Equivalence class

An equivalence class of an element $a\in A$ (with respect to an equivalence relation $\sim$ on $A$) is the set

Equivalence classes are subsets of $A$ that bundle together elements deemed “the same” by $\sim$. The collection of all equivalence classes is the quotient set $A/{\sim}=\{[a]: a\in A\}$, and it is a partition of $A$.

Examples:

• If $\sim$ is congruence modulo $3$ on $\mathbb{Z}$, then $[1]=\{\dots,-5,-2,1,4,7,\dots\}$.
• If $\sim$ is equality on a set $A$, then for each $a\in A$ one has $[a]=\{a\}$.