For all sets A and B , A c B c (A B) c
Question:
For all sets A and B, Ac ∪ Bc ⊆ (A ∪ B)c.
The following is a proposed proof for the statement.
1) Suppose A and B are any sets, such that x ∈ Ac ∪ Bc.
2) Then x ∈ Ac or x ∈ Bc 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 Ac ∪ Bc ⊆ (A ∪ B)c by definition of subset.
Identify the error(s) in the proposed proof. (Select all that apply.)
A) It is possible for x ∈ Ac ∪ Bc to be true and x ∈ Ac or x ∈ Bc to be false.
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) The proof assumes what is to be proved.
E) The proof does not handle the case when A ⊆ B.
Expert Answer:
