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How to show/prove that if A and B are sets, then (A ∩ B) ∪ (A ∩ B') = A
Problem 19.b in Section 2.2, Set Operations, Discrete Mathematics, 7th Ed Show that if A and B are sets, then (A ∩ B) ∪ (A ∩ B') = A //*Other books have it as "show (A ∩ B) ∪ (A - B) = A."*// //*(A intersect B) union (A intersect B') = A*// Solution Key definitions: A ⊂ B = {x : x ∈ A → x ∈ B} A ∩ B = {x : x ∈ A ∧ x ∈ B} A ∩ B' = A - B = {x : x ∈ A ∧ x ∉ B} A ∪ B = {x : x ∈ A ∨ x ∈ B} To show that (A ∩ B) ∪ (A ∩ B') = A, we need to show 2 things: 1. (A ∩ B) ∪ (A ∩ B') ⊂ A 2. A ⊂ (A ∩ B) ∪ (A ∩ B') //*Basically, you have 2 conditionals in a form "If P then Q" that you have to prove. To prove such conditional, you suppose that P is true, and you show that Q is true using facts about P.*// 1. Claim: (A ∩ B) ∪ (A ∩ B') ⊂ A. Proof: Suppose that x ∈ (A ∩ B) ∪ (A ∩ B'). Then, either x ∈ (A ∩ B) or x ∈ (A ∩ B'). So, either x ∈ A and x ∈ B or x ∈ A and x ∉ B. In either case, x ∈ A; thus, (A ∩ B) ∪ (A ∩ B') ⊂ A. 2. Claim: A ⊂ (A ∩ B) ∪ (A ∩ B') Proof: Suppose that x ∈ A. Then either x ∈ B or x ∉ B. If x ∈ B, then x ∈ A and x ∈ B; thus, x ∈ (A ∩ B). If x ∉ B, then x ∈ A and x ∉ B; thus, x ∈ (A ∩ B'). So, either x ∈ (A ∩ B) or x ∈ (A ∩ B'); that is, x ∈ (A ∩ B) ∪ (A ∩ B'). Therefore, A ⊂ (A ∩ B) ∪ (A ∩ B'). Q.E.D. Source(s): Grossman, Jerrold W., and Kenneth H. Rosen. "2.2 Set Operations." Student's Solutions Guide to Accompany Discrete Mathematics and Its Applications. 7th ed. New York: McGraw-Hill, 2012. 55. Print.
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