91Ó°ÊÓ

Draw the Hasse diagram for the poset \((\mathscr{P}(M), \subseteq)\), where \(q=\\{1,2,3,4\\}\)

If \(A=\\{w, x, y, z\\}\), determine the number of relations on A that are (a) reflexive; (b) symmetric; (c) reflexive and symmetric; (d) reflexive and contain \((x, y) ;\) (e) symmetric and contain \((x, y)\); (f) antisymmetric; (g) antisymmetric and contain \((x, y) ;(\mathrm{h})\) symmetric and antisymmetric; and (i) reflexive, symmetric, and antisymmetric.

How many of the equivalence relations on \(A=\) \(\\{a, b, c, d, e, f\\}\) have (a) exactly two equivalence classes of size 3 ? (b) exactly one equivalence class of size \(3 ?\) (c) one equivalence class of size \(4 ?\) (d) at least one equivalence class with three or more elements?

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 \([x] \neq[z]\)

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) \(r=23\);

a) Draw the Hasse diagram for the set of positive integer divisors of (i) 2 ; (ii) \(4 ;\) (iii) 6 ; (iv) 8 ; (v) 12; (vi) 16; (vii) 24 ; (viii) 30 ; (ix) 32 . b) For all \(2 \leq n \leq 35\), show that the Hasse diagram for the set of positive-integer divisors of \(n\) looks like one of the nine diagrams in part (a). (Ignore the numbers at the vertices and concentrate on the structure given by the vertices and edges.) What happens for \(n=36 ?\) c) For \(n \in \mathbf{Z}^{+}, \tau(n)=\) the number of positive-integer divisors of \(n .\) (See Supplementary Exercise 32 in Chapter 5.) Let \(m, n \in \mathbf{Z}^{+}\)and \(S, T\) be the sets of all positive-integer divisors of \(m, n\), respectively. The results of parts (a) and (b) imply that if the Hasse diagrams of \(S, T\) are structurally the same, then \(\tau(m)=\tau(n)\). But is the converse true? d) Show that each Hasse diagram in part (a) is a lattice if we define \(\mathrm{glb}[x, y\\}=\operatorname{gcd}(x, y)\) and lub \([x, y]=\operatorname{lcm}(x, y)\)

Let \(A=\\{1,2,3,4,5,6,7\\} .\) How many symmetric relations on \(A\) contain exactly (a) four ordered pairs? (b) five ordered pairs? (c) seven ordered pairs? (d) eight ordered pairs?

a) In how many ways can one totally order the partial order of positive- integer divisors of \(96 ?\) b) How many of the total orders in part (a) start with \(96>32 ?\) c) How many of the total orders in part (a) end with \(3>1 ?\) d) How many of the total orders in part (a) start with \(96>32\) and end with \(3>1 ?\) e) How many of the total orders in part (a) start with \(96>48>32>16 ?\)

If the complete graph \(K_{n}\) has 703 edges, how many vertices. does it have?

Find the number of ways to totally order the partial order of all positive- integer divisors of (a) 24 ; (b) \(75 ;\) and (c) 1701.