Equivalence Relation

Equivalence Relation

## Definition / Concept

A relation is an equivalence relation if it is simultaneously reflexive, symmetric and transitive.

## Formula

$$R\text{ is an equivalence relation}\Leftrightarrow R\text{ is reflexive, symmetric and transitive}$$

## Conditions / Special Cases

Reflexivity:

$$\forall a\in A,\quad aRa$$

Symmetry:

$$aRb\Rightarrow bRa$$

Transitivity:

$$aRb,\ bRc\Rightarrow aRc$$

## Terminologies

- Equivalence relation → Relation satisfying reflexivity, symmetry and transitivity.

- Equivalence class → Set of all elements related to a particular element.

## Usage

- Used to divide a set into mutually disjoint equivalence classes.