Equivalence Relation

up:: 020 MOC Mathematics

Given a set , a relation is an equivalence relation if it satisfies

1. Reflexivity:
2. Commutativity:
3. Transitivity:

All points equivalent to each other belong to the same Equivalence Class

Note that All different equivalence classes are disjoint, since having a single point in common makes them the same class.

Examples

One can prove that All pseudometric spaces induce metric spaces via the equivalence relation .

Also, a function induces an equivalence relation of the sort , partitioning its image into different “fragments” if points map to the same point.