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Reorder subtasks
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Move subtask with concrete example to the end, because the subtasks with proofs are to be done _in general_, not only for this example.
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Lipen committed Nov 8, 2023
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\item Let $R_{\theta}$ be a relation of $\theta$-similarity (clearly, $\theta \in [0; 1] \subseteq \Real$) of finite non-empty sets defined as follows: a set~$A$ is said to be \textit{$\theta$-similar} to~$B$ \textit{iff} the Jaccard index $\Jac(A,B) = \frac{\card{A \intersection B}}{\card{A \union B}}$ for these sets is at least~$\theta$, \ie $\Pair{A, B} \in R_{\theta} \iff \Jac(A,B) \geq \theta$.

\begin{subtasks}
\item Draw the graph of a relation $R_{\theta} \subseteq \Set{A_i}^2$, where $\theta = 0.25$, $A_1 = \Set{1,2,5,6}$, $A_2 = \Set{2,3,4,5,7,9}$, $A_3 = \Set{1,4,5,6}$, $A_4 = \Set{3,7,9}$, $A_5 = \Set{1,5,6,8,9}$.
\item Determine whether $\theta$-similarity is a tolerance relation.
\item Determine whether $\theta$-similarity is an equivalence relation.
\item Draw the graph of a relation $R_{\theta} \subseteq \Set{A_i}^2$, where $\theta = 0.25$, $A_1 = \Set{1,2,5,6}$, $A_2 = \Set{2,3,4,5,7,9}$, $A_3 = \Set{1,4,5,6}$, $A_4 = \Set{3,7,9}$, $A_5 = \Set{1,5,6,8,9}$.
\end{subtasks}


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