Partition of a set

Definition

Let $A$ be a set. A partition of $A$ is a set $P$ of subsets of $A$, called blocks, with these three properties:

1. each block is nonempty;
2. the blocks are pairwise disjoint;
3. the union of the blocks is $A$.

Equivalently, each element of $A$ is in exactly one block.

Remarks

The set $\{\{1, 3\}, \{2\}\}$ is a partition of $\{1, 2, 3\}$. The even and the odd integers form a partition of $\mathbb{Z}$ with two blocks. The equivalence classes form a partition, and each partition comes from exactly one equivalence relation. See the correspondence between equivalence relations and partitions.

Explanations

L1 human

A partition is a way to sort all elements into boxes. No box is empty, and each element goes into exactly one box.