Final answer:
Using Venn diagrams, we can prove that A - (B ∪ C) equals (A - B) ∩ (A - C) by showing the same area is shaded in both diagrams. The statement A×B = B×A is disproved by noting that ordered pairs in the Cartesian Product do not necessarily match when the order is reversed, unless the sets are identical or empty.
Step-by-step explanation:
Part A: Proof of Set Equality
To prove that A - (B ∪ C) = (A - B) ∩ (A - C), we will use Venn diagrams to illustrate the sets involved. A Venn diagram visually represents the elements of sets and their intersections. Here, A - (B ∪ C) means the set of elements that are in A and not in (B union C). Meanwhile, (A - B) ∩ (A - C) implies the intersection of the elements in A not in B with the elements in A not in C.
Step 1: Draw two Venn diagrams. The first shows set A and the union of sets B and C with the section A - (B ∪ C) shaded. The second diagram shows set A with separate sections shaded for A - B and A - C. The overlap of these two shaded sections represents (A - B) ∩ (A - C).
Step 2: The shaded area in both diagrams will be the same, showing the elements that are exclusively in set A and not in sets B or C. This confirms the equality A - (B ∪ C) = (A - B) ∩ (A - C).
Part B: Disproving Commutativity of Cartesian Product
For the statement A×B = B×A, we disprove it by counterexample. The Cartesian Product A×B consists of all ordered pairs (a, b) where a is in A and b is in B. Similarly, B×A consists of all ordered pairs (b, a) where b is in B and a is in A.
If A and B are not the same set, it's possible to find elements a in A and b in B such that (a, b) does not equal (b, a), disproving the statement. Therefore, the commutativity does not generally hold for the Cartesian Product of sets, unless A and B are the same set or both are empty.