Fundamental Theorem on Equivalence Relations/Examples/Arbitrary Equivalence on Set of 6 Elements 1

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Example of Use of Fundamental Theorem on Equivalence Relations

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:

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


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