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 $\RR \subset S \times S$ 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:
\(\ds \eqclass 1 \RR\) | \(=\) | \(\ds \set {1, 3, 4}\) | ||||||||||||
\(\ds \eqclass 2 \RR\) | \(=\) | \(\ds \set {2, 5, 6}\) |