Let (A, R) be a poset, and let ∅ ≠ C ⊆ A. If (C × C) ⋂ R = ∅, then for all distinct x, y ∈ C we have x R y and y R x. The elements of C are said to form an antichain in the poset (A, R).
(a) Find an antichain with three elements for the poset given in the Hasse diagram of Fig. 7.18(d). Determine a largest antichain containing the element 6. Determine a largest antichain for this poset.
(b) If U = {1, 2, 3, 4}, let A = P(U). Find two different antichains for the poset (A, ⊆). How many elements occur in a largest antichain for this poset?
(c) Prove that in any poset (A, R), the set of all maximal elements and the set of all minimal elements are antichains.