What is the correct answer to the Problem in Description below?

Question

Prove the following statement. Assume that all sets are subsets of a universal set U.

For all sets A and B, if A^c ⊆ B then A ∪ B = U.

Hint: Once you have assumed that A and B are any sets with A^c ⊆ 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.)

Answer Choices (select all that apply):

A. A ∩ B ⊆ U

B. U ⊆ A ∩ B

C. A ∪ B ⊆ U

D. U ⊆ A ∪ B

Write the proof as a free response.

Answer 1

The statement is true i.e. we have that if (complement of A) is a subset of B then (A union B)=U. Trivially since A and B are subsets of U we have that (A union B) is a subset of U.

Now let x be an element of U. Then there are two possibilities. Either x is an element of B or x is not an element of B.

In the first case we have that x belongs to B and hence, x belongs to (A union B).

In the second case, if x is not an element of B then x must be an element of A since if x is not element of A then x belongs to the complement of A which is a subset of B and hence, x belongs to B.

Thus, in both cases if x is an element of U then x is also an element of (A union B) which implies the desired equality (A union B)=U.

The proof is complete.