Mathematics 220
Homework 5
Due February 11th/12th
1. Question 3.18. Prove that 5x − 11 is even if and only if x is odd.
2. Prove that the product of two integers ab is odd if and only if both a and b are odd.
3. Prove that n3 is even only if n is even.
4. Prove that for any sets A and B, A∆B = ∅ iff A = B.
Here recall that A∆B = (A − B) ∪ (B − A).
5. Consider the statement: For any sets A and B, (A ∪ B) − B = A. If it is true, provide
a proof. If it is false give a counter-example.
6. Question 4.56 Let A, B, C be sets. Prove that (A − B) ∪ (A − C) = A − (B ∩ C).
7. Question 4.58 Let A, B, C, D be sets. Prove that (A×B)∩(C×D) = (A∩C)×(B∩D).
Note: (A × B) ∪ (C × D) = (A ∪ C) × (B ∪ D) is false: can you find a counterexample?
8. Question 4.2. Let a, b ∈ Z, a, b 6= 0. Prove that if a|b and b|a, then a = b or a = −b.
9. Question 4.4 Let x, y ∈ Z. Prove that if 3 ∤ x and 3 ∤ y then 3 | (x2 − y 2).
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