# 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 $\mathcal R \subset S \times S$ be a relation on $S$ defined as:

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

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

 $\displaystyle \eqclass 1 {\mathcal R}$ $=$ $\displaystyle \set {1, 2, 3}$ $\displaystyle \eqclass 4 {\mathcal R}$ $=$ $\displaystyle \set {4, 5}$ $\displaystyle \eqclass 6 {\mathcal R}$ $=$ $\displaystyle \set 6$

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

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

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

 $\displaystyle 1$ $\mathcal R$ $\displaystyle 3$ $\displaystyle 3$ $\mathcal R$ $\displaystyle 4$ $\displaystyle 2$ $\mathcal R$ $\displaystyle 6$ $\displaystyle \forall a \in A: \ \$ $\displaystyle \size {\eqclass a {\mathcal R} }$ $=$ $\displaystyle 3$

Then the equivalence classes of $\mathcal R$ are:

 $\displaystyle \eqclass 1 {\mathcal R}$ $=$ $\displaystyle \set {1, 3, 4}$ $\displaystyle \eqclass 2 {\mathcal R}$ $=$ $\displaystyle \set {2, 5, 6}$