Partition - Knowlpedia

Reflexive: a ∼ a a\sim a a ∼ a .
Symmetric: if a ∼ b a\sim b a ∼ b , then b ∼ a b\sim a b ∼ a .
Transitive: if a ∼ b a\sim b a ∼ b and b ∼ c b\sim c b ∼ c , then a ∼ c a\sim c a ∼ c .

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

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

Remarks

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 $\mathbb{Z}$ .