Partition

A partition of a set $X$ is a set $\mathcal{P}$ of subsets of $X$ (called blocks or parts) such that:

1. (Nonempty blocks) For every $B\in\mathcal{P}$, one has $B\neq\varnothing$.
2. (Pairwise disjoint) For all $B,C\in\mathcal{P}$, if $B\neq C$ then $B\cap C=\varnothing$.
3. (Covers $X$) $\bigcup_{B\in\mathcal{P}} B = X$.

Partitions are in bijective correspondence with equivalence relations : an equivalence relation yields a partition by its equivalence classes , and a partition yields an equivalence relation by declaring two elements equivalent exactly when they lie in the same block.

Examples:

• $\bigl\{\{1,3\},\{2\},\{4\}\bigr\}$ is a partition of $\{1,2,3,4\}$.
• The set of residue classes modulo $n$ forms a partition of $\mathbb{Z}$.