Prove the following statement. Assume that all sets are subsets of a universal set U. For all sets A and B, if  Ac ⊆ B  then  A ∪ B = U. Hint: Once you have assumed that A and B are any sets with  Ac ⊆ B,  which of the following must you show to be true in order to deduce the set equality in the conclusion of the given statement? (Select all that apply.)   A ∩ B ⊆ U U ⊆ A ∪ B A ∪ B ⊆ U U ⊆ 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.

Could you write a formal proof for the following:  For all sets A, B, and C, if A ⊆ B and B ∩ C = ∅ then A ∩ C = ∅.

Find a counterexample to show that the following statement is false.Assume that all sets are subsets of a universal set U. For all sets A, B and C,(A∪B)∩C=A∪(B∩C)