Correspondence between equivalence relations and partitions

Statement

Let $A$ be a set. Send each equivalence relation $\sim$ on $A$ to its set of equivalence classes $A/{\sim}$. This map is a bijection from the set of equivalence relations on $A$ to the set of partitions 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".