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Chapter 7: Problem 9
If the complete graph \(K_{n}\) has 45 edges, what is \(n\) ?
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Let \(p, q, r, s\) be four distinct primes and \(m, n, k, \ell \in \mathbf{Z}^{+}\) How many edges are there in the Hasse diagram of all positive divisors of \((a) p^{3} ;(b) p^{m} ;\) (c) \(p^{3} q^{2} ;\) (d) \(p^{m} q^{n} ;\) (e) \(p^{3} q^{2} r^{4}\) (f) \(p^{m^{\prime \prime}} q^{n} r^{k} ;(g) p^{3} q^{2} r^{4} s^{7} ;\) and \((\mathrm{h}) p^{m} q^{n} r^{k} s^{L} ?\)
Let \(A=\\{1,2,3,4,5,6,7\\}\), For each of the following values of \(r\), determine an equivalence relation \(\mathscr{H}\) on \(A\) with \(|\mathscr{R}|=\) \(r\), or explain why no such relation exists. (a) \(r=6 ;\) (b) \(r=7 ;\) (c) \(r=8 ;\) (d) \(r=9\); (e) \(r=11 ;\) (f) \(r=22\) (h) \(r=30\); (i) \(r=31\) (f) \(r=22 ;\) (g) \(r=23\);
If \((A, \mathscr{})\) is a poset but not a total order, and \(\emptyset \neq B \subset A\), does it follow that \((B \times B) \cap \mathscr{R}\) makes \(B\) into a poset but not a total order?
Let \(n \in \mathbf{Z}^{+}\)with \(n>1\), and let \(A\) be the set of positive integer divisors of \(n\). Define the relation \(\mathscr{A}\) on \(A\) by \(x\). \(x y\) if \(x\) (exactly) divides \(y\). Determine how many ordered pairs are in the relation \(\mathscr{R}\) when \(n\) is (a) \(10 ;\) (b) \(20 ;\) (c) 40 ; (d) \(200 ;\) (e) 210 ; and (f) 13860 .
Let \(A=\\{v, w, x, y, z\\}\). Determine the number of relations on \(A\) that are (a) reflexive and symmetric; (b) equivalence relations; (c) reflexive and symmetric but not transitive; (d) equivalence relations that determine exactly two equivalence classes; (e) equivalence relations where \(w \in[x]\); (f) equivalence relations where \(v, w \in[x]\); (g) equivalence relations where \(w \in[x]\) and \(y \in[z]\); and (h) equivalence relations where \(w \in[x], y \in[z]\), and \([x] \neq[z]\)
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