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**Formal Definition**

An **equivalence relation** $\mathcal{R}$ on a set $S$ is a relation that satisfies the following properties:

1. Reflexive: **x**$\mathcal{R}$**x** for all **x**$\in$$S$

2. Symmetric: If **x**$\mathcal{R}$**y**, then **y**$\mathcal{R}$**x**.

3. Transitive: If **x**$\mathcal{R}$**y** and **y**$\mathcal{R}$**z** then **x**$\mathcal{R}$**z**.

**Informal Definition**

The relation that holds between two elements *if and only if* they are members of the same cell within a set that has been partitioned into cells such that every element of the set is a member of one and only one cell of the partition.

**Example(s)**

Let the set **{a, b, c}** have the equivalence relation

**{**(a,a), (b,b), (c,c) (b,c), (c,b)**}**.

The following sets are equivalence classes of this relation:

[a] = **{**a**}**, [b] = [c] = **{**b,c**}**.

The set of all equivalence classes for this relation is

**{ {**a**}**, **{**b,c**} }**.

**Non-example(s)**

- The relation $\geq$ between real numbers is reflexive and transitive, but not symmetric. For example,
**7$\geq$5**does not imply that**5$\geq$7**. - The empty relation $\mathcal{R}$ on a non-empty set $X$ is symmetric but not reflexive. (If $X$ is also empty then $\mathcal{R}$ is reflexive.)

**Additional Comments**

The following are considered equivalence relations:

- "is equal to" on the set of real numbers
- "has the same birthday as" on the set of all people
- "is congruent to, modulo n" on the integers
- "has the same image under a function" on the elements of the domain of the function