Question

Question 2
Draw Hasse diagram. m n iff m divides n, A = the set of all positive-integer divisors of 36.
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Answer 1

Answer:

Upper level math

Discrete math question & answer

dravid1992sla dravid1992sla

Asked 1yr ago.

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. $$\begin{matrix} \text{ } & \text{vv} & \text{ } & \text{ωω} & \text{ }\ \text{ } & \text{0} & \text{1} & \text{0} & \text{1}\ \text{s1} & \text{s7s7} & \text{s6s6} & \text{1} & \text{0}\ \text{s2} & \text{s7s7} & \text{s7s7} & \text{0} & \text{0}\ \text{s3} & \text{s7s7} & \text{s2s2} & \text{1} & \text{0}\ \text{s4} & \text{s2s2} & \text{s3s3} & \text{0} & \text{0}\ \text{s5} & \text{s3s3} & \text{s7s7} & \text{0} & \text{0}\ \text{s6} & \text{s4s4} & \text{s1s1} & \text{0} & \text{0}\ \text{s7} & \text{s3s3} & \text{s5s5} & \text{1} & \text{0}\ \text{s8} & \text{s7s7} & \text{s3s3} & \text{0} & \text{0}\ \end{matrix}$$ b) For all 2≤n≤35,2≤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)=n∈Z+,τ(n)= the number of positive-integer divisors of n. Let m,n∈Z+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).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).

Step-by-step explanation:

A Hasse diagram is a graphical representation of a partially ordered set (poset). The set consisting of the divisors of a positive integer can be considered a poset under the relation if divides . For this poset, any edge in the diagram is such that the number below divides the number immediately above. If is a product of prime numbers, then is isomorphic to the set of subsets of with the relation if is contained in ; hence, is a Boolean algebra in this case.