---
tag: 074482Q7CD
kind: statement
role: theorem
name: Correspondence between equivalence relations and partitions
aliases: [fundamental-theorem-of-equivalence-relations]
msc: ["03E20"]
uses: [01HT2CH8BH, 0SNQEF7Z53]
sources:
  - url: https://en.wikipedia.org/wiki/Partition_of_a_set
    locator: Partitions and equivalence relations
---

## Statement

Let $A$ be a set.
Send each [[equivalence relation|0SGTMBEWRF]] $\sim$ on $A$ to its set of [[equivalence classes|082RMRSY26]] $A/{\sim}$.
This map is a [[bijection|0SNQEF7Z53]] from the set of equivalence relations on $A$ to the set of [[partitions|09CCE86A6D]] of $A$.
Its inverse sends a partition $P$ to the relation $\sim_P$ with $a \sim_P b$ when $a$ and $b$ are in the same block of $P$.

## Remarks

The relation $\sim_P$ is an equivalence relation, and its classes are the blocks of $P$.
A smaller equivalence relation, one that relates fewer pairs, gives a finer partition.
The number of partitions of a set with $n$ elements is the Bell number $B_n$. For example, $B_3 = 5$.

## Explanations

### L1 human

An equivalence relation and a partition are two descriptions of the same thing.
From the relation you get the boxes, and from the boxes you get the relation "is in the same box as".