# 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.

### [...]. Definition.

Let \( S \) be a set. A [...] of \( S \) is a family of [something something] subsets of \( S \) that [...].

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 \( \mathcal{P} \).