Fundamental Theorem on Equivalence Relations/Examples/Arbitrary Equivalence on Set of 6 Elements 2

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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 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:

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


Proof

We have that:

$1 \mathrel {\mathcal R} 3$ and $3 \mathrel {\mathcal R} 4$

As $\mathcal R$ is an equivalence relation, it follows that $\mathcal R$ is transitive and so:

$1 \mathrel {\mathcal R} 4$

Thus:

$\eqclass 1 {\mathcal R} \subseteq \set {1, 3, 4}$

but as:

$\size {\eqclass a {\mathcal R} } = 3$

it follows that:

$\eqclass 1 {\mathcal R} = \set {1, 3, 4}$


There are $6$ elements of $S$.

Thus the other $3$ elements must all be in the same equivalence class of $\mathcal R$ which does not contain $1$, for example.

Thus we have:

$\eqclass 2 {\mathcal R} \subseteq \set {2, 5, 6}$

and the information we were given that $2 \mathrel {\mathcal R} 6$ was superfluous.

$\blacksquare$


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