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 $\RR \subset S \times S$ 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:
\(\ds \eqclass 1 \RR\) | \(=\) | \(\ds \set {1, 2, 3}\) | ||||||||||||
\(\ds \eqclass 4 \RR\) | \(=\) | \(\ds \set {4, 5}\) | ||||||||||||
\(\ds \eqclass 6 \RR\) | \(=\) | \(\ds \set 6\) |
Sources
- 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.3$: Equivalence Relations