Section 1.4

• 14. a) The power set of any set includes at least the empty set, so the power set cannot be empty. Thus the empty set is not the power set of any set. b) This is the power set of {a}. c) This set has three elements. Since 3 is not a power of 2, this set cannot be the power set of any set. d) This is the power set of {a,b}.
• 18. A is the empty set or B is the empty set.
• 22. Suppose A ~= B and neither A nor B is empty. We must prove that A x B ~= B x A. Since A ~= B, either we can find an element x that is in A but not B, or vice versa. The two cases are similar, so without loss of generality, let us assume that x is in A but not in B. Also, since B is not empty, there is some element y in B. Then (x,y) is in A x B by definition, but it is not in B x A since x is not in B. Therefore A x B ~= B x A.
• 24. a) if S is an element of S, then by the defining condition for S we conclude that S is not an element of S, a contradiction. b) If S is not an element of S, then by the defining condition for S we conclude that it is not the case that S is not an element of S (otherwise S would be an elment of S), again a contradiction.