Question

Let A and B be two sets. then (A−B)∪B=A if and only if B is subset of A.

Answer 1

The statement (A−B)∪B=A is true if and only if B is a subset of A because only then, subtracting B from A and then uniting with B would result in the original set A.

Step-by-step explanation:

The question asks if (A−B)∪B=A is true if and only if B is a subset of A. To prove this, let's consider two scenarios:

1. If B is a subset of A, then every element of B is also in A. Subtracting B from A would remove those elements of B from A, leaving us with elements that are only in A and not in B. When we take the union of this result with B, we simply add back those elements of B that were in A, leading us again to the set A. Therefore, the statement is true.
2. If B is not a subset of A, there might exist elements in B that are not in A. When we form the union (A−B)∪B, we end up adding elements to A that weren't there before. Therefore, the union could potentially be a larger set than A, which means the statement would not hold in this case.

Therefore, the statement (A−B)∪B=A holds true if and only if B is a subset of A.