Question

Let A and B be sets. Prove $A \cap B = \emptyset \iff (A \times B) \cap (B \times A) = \emptyset$. (Use proof by contrapositive for both directions.)

          Let A and B be sets. Prove
$A \cap B = \emptyset \iff (A \times B) \cap (B \times A) = \emptyset$.
(Use proof by contrapositive for both directions.)

Let A and B be sets. Prove
A ∩ B = ∅ (A × B) ∩ (B × A) = ∅.
(Use proof by contrapositive for both directions.)

Added by Tara G.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

Instant Answer

Solved by Expert Moses Obasola

Step 1

Proof by contrapositive: Assume that (Ax B)n(B x A) = 0. We need to show that AnB = 0. Suppose there exists an element x in AnB. Then, x is in both A and B. Since (Ax B)n(B x A) = 0, x cannot be in both A and B at the same time. This is a contradiction, so AnB Show more…

Show all steps