Equivalence class

Example: In , if and only if meaning that have the same remainder when dived by 3. (you can write this as )

1. This relation is reflexive because for any , since
2. This relation is symmetric because for any with is the same as is the same as which is
3. The property is transitive as for any with , , . Since is divisble by3. Therefore, this implies that and The equivalence classes would be for this (the division by )

Definition: Let be a non-empty set equipped with an equivalence relation . The equivalence classes of an element is defined by and defined as:

• The reason we use equivalence relations is because you can divide the behaviors of elements in a set