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Chapter 7: Problem 20
For \(|A|=5\), how many relations \(\mathscr{\text { on }} A\) are there? How many of these relations are symmetric?
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For \(X=\\{0,1\\}\), let \(A=X \times X .\) Define the relation \(\mathscr{R}\) on
\(A\) by \((a, b) \mathscr{R}(c, d)\) if (i) \(a
A relation \(\mathscr{\text { on a set }} A\) is called irreflexive if for all \(a \in A,(a, a) \notin \mathscr{\text { . }}\) a) Give an example of a relation 9 on \(\mathbf{Z}\) where 9 is irreflexive and transitive but not symmetric. b) Let \(\mathscr{\text { be a nonempty relation on a set }} A\). Prove that if 9 satisfies any two of the following properties-irreflexive, symmetric, and transitive-then it cannot satisfy the third. c) If \(|A|=n \geq 1\), how many different relations on \(A\) are irreflexive? How many are neither reflexive nor irreflexive?
For sets \(A, B\), and \(C\) with relations \(\mathscr{R}_{1} \subseteq A \times B\) and \(\mathscr{R}_{2} \subseteq B \times C\), prove or disprove that \(\left(\Re{R}_{1} \circ \mathscr{R}_{2}\right)^{c}=\mathscr{R}_{2}^{c} \odot \mathscr{R}_{1}^{c}\).
If the complete graph \(K_{n}\) has 45 edges, what is \(n\) ?
a) Describe the structure of the Hasse diagram for a totally ordered poset \((A, M)\), where \(|A|=n \geq 1\). b) For a set \(A\) where \(|A|=n \geq 1\), how many relations on \(A\) are total orders?
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