Question

Let A, B, and C be sets. Prove or disprove the following: a.) Using at least two Venn diagrams, "prove" the statement A − (B ∩ C) = (A − B) ∪ (A − C). Explain in words why your diagrams show that the sets are equal. b.) Disprove the statement A ⨯ B = B ⨯ A.

Answer 1

To prove the statement A − (B ∩ C) = (A − B) ∪ (A − C), we can use Venn diagrams. Regarding statement b, it is true that A ⨯ B = B ⨯ A.

Step-by-step explanation:

To prove the statement A − (B ∩ C) = (A − B) ∪ (A − C), we can use Venn diagrams. Let's draw two Venn diagrams:

Venn diagram 1: A − (B ∩ C)

Venn diagram 2: (A − B) ∪ (A − C)

1. In Venn diagram 1, draw set A, then shade the region of A that overlaps with sets B and C.
2. In Venn diagram 2, draw set A, then shade the region of A that does not intersect with sets B and C.
3. Compare the shaded regions in both diagrams. You'll see that they are identical. This demonstrates that A − (B ∩ C) = (A − B) ∪ (A − C).

Thus, using Venn diagrams, we have proven that A − (B ∩ C) = (A − B) ∪ (A − C).

Regarding statement b, it is true that A ⨯ B = B ⨯ A. The Cartesian product A ⨯ B represents all possible ordered pairs where the first element comes from set A and the second element comes from set B. Since order does not matter in the Cartesian product, swapping A and B does not change the result. Therefore, A ⨯ B = B ⨯ A.