Fundamental Theorem on Equivalence Relations/Examples

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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:

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