5. For any set X, let P(X) denote the power set of X. Prove or disprove the following statements (In other words, determine whether each of the following statements is true or false, and prove your claim. If the statement is true, give a proof. If the statement is false, prove that the negation is true, i.e. give a counterexample). (a) For any sets A and B, P(AN B) = P(A) N P(B). (b) For any sets A and B, P(AUB) = P(A) U P(B). (c) For any sets A, B, and C, A\ (BUC)= (A\B)U (A\ C). %3D

Give formal well-written proofs when you are proving a result or a counter example otherwise Prove or disprove each of the following statements. For all sets A and B, if A n B = B, then B subset A.

Consider the following statement. For all sets A and B, (AUB)C=AC U Bº. Which sentence below expresses what it means for the statement to be false? (Select all that apply.) For all sets A and B, (A U B)C + AC U BC. There is a set A such that for all sets B, (A U B)C + AC U BC. There are sets A and B such that (A U B)C ‡ AC U BC. For all sets A, there is a set B such that (A U B)C # AC U BC. Find subsets of {1, 2, 3, 4, 5} which can be used to show that the given statement is false. (Enter the sets as A, B in a comma-separated list. Use set-roster notation or write EMPTY or for the empty set.) A, B =

6. The following statements about sets are false. For each statement, give an example, i.e., a choice of sets, for which the statement is false. Such examples are called counterexamples. They are examples that are counter to, i.e., contrary to, the assertion. (a) AUBCAn B for all A, B. (b) ANØ: = A for all A. (c) AN (BUC) = (A ^ B) UC for all A, B, C.