# Equivalence relations

### The datum of an equivalence relation is a partition

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 a
*partition* of \( S \). Compare to the case of the product of two sets, which has a set of ordered pairs as its datum.

### Partition. Definition.

Let \( S \) be a set. A partiton \( \mathcal{P} \) of \( S \) is a family of disjoint non-empty subsets of \( S \) that union to \( S \).

For example, \( \{\{1,3,5\}, \{2,4,6\}, \{0\}, \{7\}, \{8, 9\} \} \) is a partition 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 \( \mathcal{P} \).

### Quotients

The *quotient* of a set \( S \) with respect to an
equivalence relation \( \sim \) is preciely the partition \( \mathcal{P}_{\sim} \)
(the set of equivalence classes formed by \( \sim \) with respect to \( S \)). We write
the quotient as \( S \):

#### Quotients and equivalence relations, a perspective

The equivalence relation becomes equality in the quotient. Taking a quotient turns an equivalence relation into an equality.

### Example

Let \( \sim \) be the relation defined for \( \mathbb{Z} \) like so:

Then the quotient is the set containing two equivalence classes, \( \mathbb{Z}/ \mathord{\sim} = \{ [0]_{\sim}, [1]_{\sim} \} \).