Equivalence class

Definition

Let $\sim$ be an equivalence relation 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.

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.