The answer to exercise marked [BB] can be found in the Back of the Book.Suppose that (A1,≤...

The answer to exercise marked [BB] can be found in the Back of the Book.

Suppose that (A1,≤ 1) and (A2, ≤2) are partial orders.

(a) Show that the definition

for (x1,x2),(y1, y2) ∈ A1 × A2makes A1× A2a partially ordered set.

(b) Let A1 = A2 = {2, 3, 4}. Assign to A1 the partial order ≤ and to A2 the partial order |. Partially order A1 × A2 as in part (a). Show all relationships of the from (x1, x2)

(c) Draw the Hasse diagram for the partial order in (b).

(d) Find any maximal, minimal, maximum, and minimum elements that may exist in the partial order of (b).

(e) With A1 and A2as in part (b), find the glbs and lubs that exist for each of the following pairs of elements.

i. (2, 2), (3, 3)

ii. (4, 2), (3, 4)

iii. (3, 2), (2, 4)

iv. (3, 2), (3, 4)

(f) Show, by example, that if (A1,≤1) and (A2,≤2)are total orders then (A1 × A2, ≤)need not be a total order.