Dr. A. Betten
Fall 2009
MATH 281A1 Introduction to Mathematical Reasoning
Assignment # 2
Problem # 5
a) Let A and B be sets. Prove that A ∪ B = A if and only if B ⊆ A.
b) Prove that (A \ C) ∩ (B \ C) = (A ∩ B) \ C for all sets A, B, C.
Problem # 6
Which if the following statements are true and which are false? Give proofs or counterexamples.
a) For any sets A, B, C we have
A ∪ (C ∩ C) = (A ∪ B) ∩ (A ∪ C).
b) For any sets A, B, C we have
(A \ B) \ C = A − (B \ C).
c) For any sets A, B, C we have
(A \ B) ∪ (B \ C) ∪ (C \ A) = A ∪ B ∪ C.
Problem # 7
∞
Work out ∪∞
n=1 An and ∩n=1 An for the following sets
a) An = {x ∈ R | x > n} √
b) An = {x ∈ R | n1 < x < 2 + n1 }
c) An = {x ∈ R | −n < x < n1 }
√
√
d) An = {x ∈ R | 2 − n1 < x < 2 + n1 }
Problem # 8
Book 5.19 page 49
Problem # 9
Book 5.20 page 49