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Use an element argument to prove the statement. (Assume that all sets are subsets of a universal set U.) Statement: For all sets A, B, and C, (A − B) ∩ (C − B) = (A ∩ C) − B. Proof: Suppose A, B, and C are any sets. [To show that  (A − B) ∩ (C − B) = (A ∩ C) − B,  we must show that  (A − B) ∩ (C − B) ⊆ (A ∩ C) − B  and that  (A ∩ C) − B ⊆ (A − B) ∩ (C − B).] Part 1: Proof that  (A − B) ∩ (C − B) ⊆ (A ∩ C) − B Consider the sentences in the following scrambled list. By definition of set difference, x ∈ A and x ∉ B and x ∈ C and x ∉ B. By definition of intersection, x ∈ A and x ∉ B and x ∈ C and x ∉ B. By definition of set difference, x ∈ A − B and x ∈ C − B. Thus x ∈ A ∩ C by definition of intersection and the fact that x ∉ B. Therefore x ∈ (A ∩ C) − B by the definition of set difference. By definition of intersection, x ∈ A − B and x ∈ C − B. We prove part 1 by selecting appropriate statements from the list and putting them in the correct order. Suppose x ∈ (A − B) ∩ (C − B).

Use an element argument to prove the statement. (Assume that all sets are subsets of a universal set U.) Statement: For all sets A, B, and C, (A − B) ∩ (C − B) = (A ∩ C) − B. Proof: Suppose A, B, and C are any sets. [To show that  (A − B) ∩ (C − B) = (A ∩ C) − B,  we must show that  (A − B) ∩ (C − B) ⊆ (A ∩ C) − B  and that  (A ∩ C) − B ⊆ (A − B) ∩ (C − B).] Part 1: Proof that  (A − B) ∩ (C − B) ⊆ (A ∩ C) − B Consider the sentences in the following scrambled list. By definition of set difference, x ∈ A and x ∉ B and x ∈ C and x ∉ B. By definition of intersection, x ∈ A and x ∉ B and x ∈ C and x ∉ B. By definition of set difference, x ∈ A − B and x ∈ C − B. Thus x ∈ A ∩ C by definition of intersection and the fact that x ∉ B. Therefore x ∈ (A ∩ C) − B by the definition of set difference. By definition of intersection, x ∈ A − B and x ∈ C − B. We prove part 1 by selecting appropriate statements from the list and putting them in the correct order. Suppose x ∈ (A − B) ∩ (C − B).

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.1: Sets
Problem 1TFE: True or False Label each of the following statements as either true or false. Two sets are equal if...
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Use an element argument to prove the statement. (Assume that all sets are subsets of a universal set U.)
Statement: For all sets A, B, and C,
(A − B) ∩ (C − B) = (A ∩ C) − B.
Proof:
Suppose A, B, and C are any sets. [To show that 
(A − B) ∩ (C − B) = (A ∩ C) − B,
 we must show that 
(A − B) ∩ (C − B) ⊆ (A ∩ C) − B
 and that 
(A ∩ C) − B ⊆ (A − B) ∩ (C − B).]
Part 1: Proof that 
(A − B) ∩ (C − B) ⊆ (A ∩ C) − B
Consider the sentences in the following scrambled list.
  • By definition of set difference, x ∈ A and x ∉ B and x ∈ C and x ∉ B.
  • By definition of intersection, x ∈ A and x ∉ B and x ∈ C and x ∉ B.
  • By definition of set difference, x ∈ A − B and x ∈ C − B.
  • Thus x ∈ A ∩ C by definition of intersection and the fact that x ∉ B.
  • Therefore x ∈ (A ∩ C) − B by the definition of set difference.
  • By definition of intersection, x ∈ A − B and x ∈ C − B.
We prove part 1 by selecting appropriate statements from the list and putting them in the correct order.
  1. Suppose x ∈ (A − B) ∩ (C − B).
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