Question

Prove the following statement: For all sets A and B, if \(A^{c} \subseteq B\), then \(A \cup B = U\).

a) True
b) False

Answer 1

If the complement of set A is a subset of set B, then the union of A and B is the universal set U, thereby validating the statement as true.

Step-by-step explanation:

The statement we're asked to prove is: "For all sets A and B, if Ac ⊆ B, then A ∪ B = U." Here, Ac represents the complement of A, ⊆ denotes the subset relation, ∪ stands for the union of two sets, and U is the universal set.

To prove this, consider that if Ac ⊆ B, every element not in A is in B. Therefore, combining all the elements in A with all the elements in B (which include all the elements not in A due to the subset relation) would indeed cover every element in the universal set U. Hence, A ∪ B = U.

Therefore, the statement is true.