Equivalence relation

An equivalence relation is a binary relation on a set that groups elements into categories^[1] in which all members are considered "equivalent" under some criterion.

Definition

A relation $\text{[math]}$ on set $\text{[math]}$ is a equivalence relation if it satisfies the following properties:

Reflexivity: $\text{[math]}$ , $\text{[math]}$ .
Symmetry: $\text{[math]}$ such that $\text{[math]}$ , $\text{[math]}$ .
Transitivity: $\text{[math]}$ , if $\text{[math]}$ and $\text{[math]}$ , then $\text{[math]}$ .

When $\text{[math]}$ , then we say that " $\text{[math]}$ is equivalent to $\text{[math]}$ " under the relation $\text{[math]}$ .

Notes

↑ Not to be confused with category in category theory.

[1] Not to be confused with category in category theory.

[1]