1.1 – Sets and Functions Exercise: 1 Section 1.1 Question: Let U = {n ∈ N | n ≤ 10} and consider the subsets A = {1, 3, 5, 7, 9}, B = {1, 2, 3, 4, 5}, and C = {1, 2, 5, 7, 8}. Calculate the following operations. a) A ∩ B b) (B ∪ C) − A c) (A ∩ B) ∩ (A ∩ C) ∩ (B ∩ C) d) ((A − B) − C) ∩ (A − (B − C)) Solution: We apply the definitions of set operations: a) A ∩ B = {1, 3, 5} b) (B ∪ C) − A = {1, 2, 3, 4, 5, 7, 8} − {1, 3, 5, 7, 9} = {2, 4, 8} c) A ∩ B∩A ∩ C∩B ∩ C = {1, 3, 5}∩{1, 5, 7}∩{1, 2, 5} = {2, 4, 6, 7, 8, 9, 10}∩{2, 3, 4, 6, 8, 9, 10}∩{4, 6, 8, 9, 10} d) ((A − B) − C) ∩ (A − (B − C)) = ({7, 9} − C) ∩ (A − {3, 4}) = {9} ∩ {1, 5, 7, 9} = {9} Exercise: 2 Section 1.1 Question: Let U = {a, b, c, d, e, f, g} and consider the subsets A = {a, b, d}, B = {b, c, e}, and C = {c, d, f}. Calculate the following operations. a) C ∩ (A ∪ B) b) (A ∪ C) − B c) (A ∪ B ∪ C) − (A ∩ B ∩ C) d) (A − B) ∪ (B − C) Solution: We apply the definitions of set operations: a) C ∩ (A ∪ B) = C ∩ {a, b, c, d, e} = {c, d} b) (A ∪ C) − B = {a, b, c, d, f} − B = {a, d, f} c) (A ∪ B ∪ C) − (A ∩ B ∩ C) = {a, b, c, d, e, f} − ∅ = {a, b, c, d, e, f} d) (A − B) ∪ (B − C) = {a, d} ∪ {b, e} = {a, b, d, e} Exercise: 3 Section 1.1 Question: As subsets of the reals, describe the differences between the sets {3, 5}, [3, 5] and (3, 5). Solution: The set {3, 5} contains the integers 3 and 5. The closed interval [3, 5] contains all real numbers between 3 and 5 including 3 and 5, while the open interval (3, 5) contains all real numbers between 3 and 5 not including 3 and 5. Exercise: 4 Section 1.1 Question: Prove that the following are true for all sets A and B. a) A ∩ B ⊆ A. b) A ⊆ A ∪ B. Solution: We use the definitions