Equivalence relation

Definition

Let $A$ be a set. An equivalence relation on $A$ is a binary relation $\sim$ on $A$ that is reflexive, symmetric and transitive. For all $a, b, c \in A$:

1. $a \sim a$;
2. $a \sim b$ implies $b \sim a$;
3. $a \sim b$ and $b \sim c$ imply $a \sim c$.

Remarks

For a fixed integer $n \ge 1$, congruence modulo $n$ on $\mathbb{Z}$ is an equivalence relation: $a \sim b$ when $n$ divides $a - b$. An equivalence relation cuts the set into equivalence classes, and these form a partition. Each function gives an equivalence relation. See equivalence relation induced by a function.

Explanations

L1 human

An equivalence relation says when two objects are "the same for our purpose". Two integers are congruent modulo $2$ when they have the same parity. Then all even numbers count as one kind, and all odd numbers as another.