## 1.3. Equivalence Relations and Classes

### Definition 1.3.1: Equivalence Relation

Let

**S**be a set and*r*a relation between**S**and itself. We call*r*an**equivalence relation**on**S**if*r*has the following three properties:**Reflexivity**: Every element of**S**is related to itself**Symmetry**: If*s*is related to*t*then*t*is related to*s***Transitivity**: If*s*is related to*t*and*t*is related to*u*, then*s*is related to*u*.