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D. Let A and B be sets. Prove that (A × B) ^ (B × A) = Ø if and only if AnB = Ø. (Use any appropriate proof technique.)

D. Let A and B be sets. Prove that (A × B) ^ (B × A) = Ø if and only if AnB = Ø. (Use any appropriate proof technique.)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please solve the following discrete math question 

**Problem D:** Let \( A \) and \( B \) be sets. Prove that \( (A \times B) \cap (B \times A) = \emptyset \) if and only if \( A \cap B = \emptyset \). *(Use any appropriate proof technique.)* **Explanation:** - *Sets \( A \) and \( B \)*: These are any given sets without specific elements defined in the problem. - *Cartesian Product \( A \times B \)*: This is the set of all ordered pairs where the first element is from \( A \) and the second element is from \( B \). - *Intersection \( \cap \)*: This indicates the set of elements that are common to both sets involved. - *Empty Set \( \emptyset \)*: This represents a set that contains no elements. **Task:** Prove the equivalence between the Cartesian product intersection being empty and the intersection of the sets being empty, using an appropriate proof technique such as direct proof, proof by contradiction, or contraposition.
Transcribed Image Text:**Problem D:** Let \( A \) and \( B \) be sets. Prove that \( (A \times B) \cap (B \times A) = \emptyset \) if and only if \( A \cap B = \emptyset \). *(Use any appropriate proof technique.)* **Explanation:** - *Sets \( A \) and \( B \)*: These are any given sets without specific elements defined in the problem. - *Cartesian Product \( A \times B \)*: This is the set of all ordered pairs where the first element is from \( A \) and the second element is from \( B \). - *Intersection \( \cap \)*: This indicates the set of elements that are common to both sets involved. - *Empty Set \( \emptyset \)*: This represents a set that contains no elements. **Task:** Prove the equivalence between the Cartesian product intersection being empty and the intersection of the sets being empty, using an appropriate proof technique such as direct proof, proof by contradiction, or contraposition.
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