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Extracted Problem:

Let $A, B, C, D$ be subsets of a set $E$. Analyze the following statements:

1. $(A \times C) \cup (B \times D) = (A \cup B) \times (C \cup D)$
2. $(A \times E) \cap (B \times E) = (A \cap B) \times E$

The question asks which of the following is not correct:

• A. 1) true, 2) true
• B. 1) true, 2) false
• C. 1) false, 2) true
• D. 1) false, 2) false

Analysis:

Statement 1:

$(A \times C) \cup (B \times D) \stackrel{?}{=} (A \cup B) \times (C \cup D)$

Using the Cartesian product and union rules, this statement is correct because: $\text{The union of Cartesian products distributes over the union of sets.}$

Statement 2:

$(A \times E) \cap (B \times E) \stackrel{?}{=} (A \cap B) \times E$

Using the Cartesian product and intersection rules, this statement is also correct because: $\text{The intersection of Cartesian products distributes over the intersection of sets.}$

Answer:

Both statements are true. Therefore, the correct option is:

• A. 1) true, 2) true

Would you like a more detailed breakdown of any part of this explanation? Here are some related questions to consider:

1. How are Cartesian products defined mathematically?
2. What are the main properties of union and intersection in set theory?
3. Can you provide a counterexample where these properties fail?
4. How can Venn diagrams help visualize Cartesian product relationships?
5. Why does the Cartesian product behave distributively with respect to union and intersection?

Tip: Always verify set identities by substituting simple sets to check the relationships systematically.