Proof that (A ∩ C) − B ⊆ (A − B) ∩ (C − B) Consider the sentences in the following scrambled list. By definition of intersection x ∈ A ∩ C and x ∉ B. By definition of set difference x ∈ A ∩ C and x ∉ B. By definition of intersection, x ∈ (A − B) ∩ (C − B). By definition of set difference, x ∈ A and x ∈ C. So by definition of set difference, x ∈ A − B and x ∈ C − B. By definition of intersection, x ∈ A and x ∈ C. Hence both x ∈ A and x ∉ B and also x ∈ C, and x ∉ B. We prove part 2 by selecting appropriate sentences from the list and putting them in the correct order. Suppose x ∈ (A ∩ C) − B.
Proof that (A ∩ C) − B ⊆ (A − B) ∩ (C − B) Consider the sentences in the following scrambled list. By definition of intersection x ∈ A ∩ C and x ∉ B. By definition of set difference x ∈ A ∩ C and x ∉ B. By definition of intersection, x ∈ (A − B) ∩ (C − B). By definition of set difference, x ∈ A and x ∈ C. So by definition of set difference, x ∈ A − B and x ∈ C − B. By definition of intersection, x ∈ A and x ∈ C. Hence both x ∈ A and x ∉ B and also x ∈ C, and x ∉ B. We prove part 2 by selecting appropriate sentences from the list and putting them in the correct order. Suppose x ∈ (A ∩ C) − B.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 15E: Let A=R0, the set of all nonzero real numbers, and consider the following relations on AA. Decide in...
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Proof that (A ∩ C) − B ⊆ (A − B) ∩ (C − B)
Consider the sentences in the following scrambled list.
By definition of intersection x ∈ A ∩ C and x ∉ B.
By definition of set difference x ∈ A ∩ C and x ∉ B.
By definition of intersection, x ∈ (A − B) ∩ (C − B).
By definition of set difference, x ∈ A and x ∈ C.
So by definition of set difference, x ∈ A − B and x ∈ C − B.
By definition of intersection, x ∈ A and x ∈ C.
Hence both x ∈ A and x ∉ B and also x ∈ C, and x ∉ B.
We prove part 2 by selecting appropriate sentences from the list and putting them in the correct order.
Suppose x ∈ (A ∩ C) − B.
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