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Add task about equivalence relations and classes
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Lipen committed Oct 1, 2024
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transitive closure of~$R$ is not transitive.


% Task: Explore the equivalence classes.
\item Let R be the relation on the set of all colorings of the $2 \times 2$ checkerboard where each of the four squares is colored either \textit{red} or \textit{blue} so that $\Pair{C_1, C_2}$, where $C_1$ and~$C_2$ are $2 \times 2$ checkerboards with each of their four squares colored blue or red, belongs to~$R$ if and only if $C_2$ can be obtained from~$C_1$ either by rotating the checkerboard or by rotating it and then reflecting it.
\begin{subtasks}
\item Show that $R$ is an equivalence relation.
\item What are the equivalence classes of $R$?
\end{subtasks}


% Task: Explore the inverse of a composition.
\item Consider two relations $R \subseteq A \times B$ and $S \subseteq B \times C$.
Prove that $(S \circ R)^{-1} = R^{-1} \circ S^{-1}$.
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