Equivalence Class

Equivalence Class

## Definition / Concept

For an equivalence relation $R$ on $A$, the equivalence class of $a\in A$ contains all elements related to $a$.

## Formula

$$[a]=\{x\in A:xRa\}$$

## Conditions / Special Cases

$$a\in[a]$$

If:

$$aRb$$

then:

$$[a]=[b]$$

If:

$$a\not Rb$$

then:

$$[a]\cap[b]=\varnothing$$

Equivalence classes form a partition of $A$:

$$A=[a_1]\cup[a_2]\cup\cdots\cup[a_k]$$

## Usage

- Used to group elements satisfying the same equivalence condition.

- Commonly used in congruence and divisibility-based problems.