Relations

Let be a set. A relation on is a subset of . If is a relation on , we define the statement for every .

Let be a relation on a set . We define the following properties of :

• Reflexivity: for every
• Symmetry: for every
• Transitivity: for every

If is reflexive, symmetric, and transitive, we call it an equivalence relation.

Let be an equivalence relation on a set , and let . Then is called an equivalence class with respect to if for every and for every .

denotes the equivalence class of , and contains every such that .

Let be an equivalence relation on a set . Then every element is contained in one and only one equivalence class of with respect to , namely .

Proof:
Suppose is an equivalence class of with respect to such that . Then we have , so .