---
tag: 09CCE86A6D
kind: definition
name: Partition of a set
msc: ["03E20"]
defines: 08QMKTWF9T
uses: [0FE8Q5S42Y, 0W4RPA8QH3, 05GK32H72H]
sources:
  - url: https://en.wikipedia.org/wiki/Partition_of_a_set
    locator: lead section
---

## Definition

Let $A$ be a set.
A partition of $A$ is a set $P$ of [[subsets|0FE8Q5S42Y]] of $A$, called blocks, with these three properties:

1. each block is nonempty;
2. the blocks are pairwise [[disjoint|0W4RPA8QH3]];
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|01HT2CH8BH]], and each partition comes from exactly one equivalence relation. See [[the correspondence between equivalence relations and partitions|074482Q7CD]].

## 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.