(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 ≤ n ≤ 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 ∈ Z+, τ(n) = the number of positive-integer divisors of n. (See Supplementary Exercise 32 in Chapter 5.) Let m, n ∈ 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 τ(m) = τ(n). But is the converse true?
(d) Show that each Hasse diagram in part (a) is a lattice if we define glb{x, y} = gcd(x, y) and lub{x, y} = lcm(x, y).