# Fundamental Theorem on Equivalence Relations/Examples

## Examples of Use of Fundamental Theorem on Equivalence Relations

### Arbitrary Equivalence on Set of $6$ Elements: $1$

Let $S = \set {1, 2, 3, 4, 5, 6}$.

Let $\RR \subset S \times S$ be a relation on $S$ defined as:

$\RR = \set {\tuple {1, 1}, \tuple {1, 2}, \tuple {1, 3}, \tuple {2, 1}, \tuple {2, 2}, \tuple {2, 3}, \tuple {3, 1}, \tuple {3, 2}, \tuple {3, 3}, \tuple {4, 4}, \tuple {4, 5}, \tuple {5, 4}, \tuple {5, 5}, \tuple {6, 6} }$

Then $\RR$ is an equivalence relation which partitions $S$ into:

 $\ds \eqclass 1 \RR$ $=$ $\ds \set {1, 2, 3}$ $\ds \eqclass 4 \RR$ $=$ $\ds \set {4, 5}$ $\ds \eqclass 6 \RR$ $=$ $\ds \set 6$

### Arbitrary Equivalence on Set of $6$ Elements: $2$

Let $S = \set {1, 2, 3, 4, 5, 6}$.

Let $\RR \subset S \times S$ be an equivalence relation on $S$ with the properties:

 $\ds 1$ $\RR$ $\ds 3$ $\ds 3$ $\RR$ $\ds 4$ $\ds 2$ $\RR$ $\ds 6$ $\ds \forall a \in A: \,$ $\ds \size {\eqclass a \RR}$ $=$ $\ds 3$

Then the equivalence classes of $\RR$ are:

 $\ds \eqclass 1 \RR$ $=$ $\ds \set {1, 3, 4}$ $\ds \eqclass 2 \RR$ $=$ $\ds \set {2, 5, 6}$