product—that is, prove that |x| |y| = |x y| for any real numbers x and y. 1.Suppose that x, y ∈ R satisfy |x| ≤ |y|. Prove that |x+y| 2 ≤ |y|. 2.Let A and B be sets. Prove that A × B = B × A if and...

product—that is, prove that |x| |y| = |x y| for any real numbers x and y.

1.Suppose that x, y ∈ R satisfy |x| ≤ |y|. Prove that |x+y| 2 ≤ |y|.

2.Let A and B be sets. Prove that A × B = B × A if and only if A = ∅ or B = ∅ or A = B. Prove the result by mutual implication, where the proof of the ⇐ direction proceeds by contrapositive.