Equivalence classes form a partition

Statement

Let $\sim$ be an equivalence relation on a set $A$. Then the set $\{[a] \mid a \in A\}$ of equivalence classes is a partition of $A$.

Remarks

Each class $[a]$ is nonempty, because $a \in [a]$ by reflexivity. If $[a]$ and $[b]$ have a common element, then $a \sim b$ by symmetry and transitivity, so $[a] = [b]$. The left cosets of a subgroup of a group form such a partition. This fact is the key step in Lagrange's theorem.

Explanations

L1 human

The three properties of an equivalence relation are exactly what you need to sort elements into classes. Reflexivity puts each element in some class. Symmetry and transitivity stop two classes from overlapping.