I have a few questions that I've done and I don't have the solutions for them so I was wondering if anyone might be able to verify my answers and let me know if there are any mistakes! Thanks a lot!
- Let V = {a,b,c,d} (where a,b,c,d are all distinct) and let A = {a,b}. I need to calculate the following:
(a) {X ⊆ V : A ⊥ X} (where ⊥ means A and X are disjoint) Attempt at solution: { ∅, {c}, {d}, {c,d}}
(b) {X ⊆ V : V ⊆ X} Attempt at solution: {{a,b,c,d}}
(c) {X ⊆ V : X ⊥ V} Attempt at solution: ∅
(d) {X ⊆ V : |X| = 1} Attempt at solution: { {a}, {b}, {c}, {d} }
(e) {X ⊆ V : |X| < 2} Attempt at solution: { ∅, {a}, {b}, {c}, {d} }
(f) {X ⊆ V : |X - A| > |X ∩ A|} Attempt at solution: { {c}, {d}, {c,d}, {c,d,a}, {c,d,b} }
- (a) If Pow(X) || Pow(Y), then X || Y. (where '||' means the sets are incomparable, meaning that neither set is a subset of the other and Pow(X) is the Power set of the set X) Attempt at solution: True
(b) Each non-empty subset of a non-empty set X is a member of some partition of X. Attempt at solution: True
(c) If X is non-empty, X ⊆ Y and {V,W} partitions Y, then {V ∩ X, W ∩ X} partitions X. Attempt at solution: False