---
tag: 082RMRSY26
kind: definition
name: Equivalence class
msc: ["03E20"]
defines: 0W95W33CMP
uses: [0SGTMBEWRF, 0TMZ89Q5KA]
sources:
  - url: https://en.wikipedia.org/wiki/Equivalence_class
    locator: lead section
---

## Definition

Let $\sim$ be an [[equivalence relation|0SGTMBEWRF]] on a set $A$ and let $a \in A$.
The equivalence class of $a$ is the set
$$[a] = \{x \in A \mid x \sim a\}.$$
Each element of $[a]$ is a representative of the class.

## Remarks

For congruence modulo $3$ on $\mathbb{Z}$, the class of $1$ is $[1] = \{\dots, -5, -2, 1, 4, 7, \dots\}$.
For $a, b \in A$, $[a] = [b]$ if and only if $a \sim b$. Otherwise, $[a]$ and $[b]$ are disjoint.
The set of all classes is the [[quotient set|0ZVHXDSZC7]].

## Explanations

### L1 human

An equivalence class collects all the objects that the relation treats as the same.
For "has the same birthday as", the class of a person is the group of all people born on the same day of the year.