If A and B are sets, then A ∩ (B − A) = ∅. Please prove this by contradiction.
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Using set builder notation, prove that A ∩ (B − A) = ∅.
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The following are two proofs that for all sets A and B, A - B ⊆ A. The first is less formal, and the second is more formal. Fill in the blanks. a) Proof: Suppose A and B are any sets. To show that A - B ⊆ A, we must show that every element in A - B is in A. But any element in A - B is in A and not in B (by definition of A - B). In particular, such an element is in A. b) Proof: Suppose A and B are any sets and x ∈ A - B. We must show that x ∈ A. By definition of set difference, x ∈ A and x ∉ B. In particular, x ∈ A [which is what was to be shown]. a) Proof option choices: 1st blank choices: a) B b) A - B c) A ∪ B d) A ∩ B 2nd blank choices: a) A b) A - B c) A ∪ B d) A ∩ B 3rd blank choice: a) A b) B c) A - B d) A ∪ B 4th blank choice: a) A b) B c) A ∪ B d) A ∩ B b) Proof option choices: 1st option blank: a) x ∈ A b) x ∈ B c) x ∈ B - A d) x ∈ A ∩ B 2nd option blank: a) x ∈ A b) x ∈ B c) x ∈ A ∩ B d) x ∈ A ∪ B 3rd option blank: a) x ∈ A b) x ∈ B c) x ∈ A ∩ B d) x ∈ A ∪ B 4th option blank: a) x ∈ A ∪ B b) x ∈ A ∩ B c) x ∈ A d) x ∈ B
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