Equivalence relations

The datum of an equivalence relation is a [...]

What is the information that is contained by an equivalence relation on a set? 

The datum of an equivalence relation on a set $S$ turns out to be [something of $S$]. Compare to the case of the product of two sets, which has a set of ordered pairs as its datum.

For example, $\{\{1,3,5\},\{2,4,6\},\{0\},\{7\},\{8,9\}\}$ is a [...] of the set $\{0,1,2,3,4,5,6,7,8,9\}$.

From equivalence relations to partitions, and back

There is a 1-1 correspondence between partitions and equivalence relations, and in this sense they represent the same notion.

Here is the process of obtaining a partition from an equivalence relation. Let $\sim$ be an equivalence relation on a set $S$. For every $a\in S$ the equivalence class of $a$ is the subset of $S$ defined by

The set of all equivalence classes is a partition of $S$, denoted ${\mathcal{P}}_{\sim }$.

Conversely, given a partition $\mathcal{P}$, we can form an equivalence relation $\sim$ where $a\sim b$ is true if $a$ and $b$ are in the same element of $\mathcal{P}$.