Consider the following statement. For all sets A and B, Ac ∪ Bc ⊆ (A ∪ B)c. Identify the error(s) in the proposed proof. (Select all that apply.)

Question

The following is a proposed proof for the statement.

1. Suppose A and B are any sets, such that x ∈ A^c ∪ B^c.
2. Then x ∈ A^c or x ∈ B^c by definition of union.
3. It follows that x ∉ A or x ∉ B by definition of complement, and so x ∉ A ∪ B by definition of union.
4. Thus x ∈ (A ∪ B)^c by definition of complement, and hence A^c ∪ B^c ⊆ (A ∪ B)^c by definition of subset.

Identify the error(s) in the proposed proof. (Select all that apply.)

Answer Choices:

A. The proof assumes what is to be proved.

B. It is possible for x ∉ A or x ∉ B to be true and x ∉ A ∪ B to be false.

C. The proof does not handle the case when B ⊆ A.

D. It is possible for x ∈ A^c ∪ B^c to be true and x ∈ A^c or x ∈ B^c to be false.

E. The proof does not handle the case when A ⊆ B.

Answer 1

B. Since , either , , or and . Suppose . Then . Either or . Suppose . Thus, . That is, it's possible that to be false.

Going one step further, we just showed that there exists an element such that , but ; that is, we showed that is not a subset of . So, the original statement is false and can't actually be proved, if that makes any sense.

You can prove, however, that ; in fact, you can prove the reverse containment as well: , which means that .