Math Proving question

Math 8 Homework 5 Spring, 2018

Due: Monday, April 16th at the start of class

1. (Velleman Problem 3.2.3 ) Suppose A ⊆ C, and B and C are disjoint. Prove that if x ∈ A then x 6∈ B.

2. Direct or contrapositive? Prove each of the following statements either directly or by contraposi- tive. Please write very clear and careful solutions, and, in particular, identify explicitly whether your proof is direct or by contrapositive. In each of the following, a, b and c are integers.

(a) If a2 doesn’t divide b2, then a doesn’t divide b.

(b) If a and b are odd then a2 + 3b2 + ab is odd.

(c) If b3 − 1 is odd, then b is even. (d) If a2

∣∣ b and b3 ∣∣ c, then a6 ∣∣ c. 3. Prove the following statements about the sets A, B and C. Within your arguments, you may find

it useful to argue directly and/or to use the contrapositive form. Remember that to show that two sets are equal, we show that each one is a subset of the other; also remember that to show that S is a subset of T , we argue “let x ∈ S . . . argue, argue, . . . Aha! Therefore x ∈ T .”

(a) A∩ (B \C) = (A∩B) \ (A∩C) (b) (A∪B) \ (A∩B) = (A\B) ∪ (B \A)

4. It is not true that for any sets A and B we have A\ (A\B) = B.

(a) Explain why the above statement is false by explicitly writing down two sets, A and B for which the equality fails.

(b) Is either of the statements A\ (A\B) ⊂ B or B ⊂ A\ (A\B) true for all sets A and B? (c) What simple set (in terms of A, B, their intersections, unions and so on) should replace B on

the right side of the equality? Prove your assertion.

Credits: Velleman refers to Daniel J. Velleman’s How To Prove It, Second Edition.