Homework 5 Solutions

# Homework 5 Solutions - HOMEWORK 5 COMMENTS 1 Non-book...

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Unformatted text preview: HOMEWORK 5: COMMENTS 1. Non-book problems (1) We’ll show that ¬ n _ i =1 P i = n ^ i =1 ( ¬ P i ) for n ≥ 2 by induction on n . The proof of the other law is obtained by swapping ∨ and ∧ everywhere. The base case is n = 2 and reads ¬ ( P 1 ∨ P 2 ) = ¬ P 1 ∧¬ P 2 . We have shown this in Homework 1. Assume as our inductive hypothesis that ¬ k _ i =1 P i = k ^ i =1 ( ¬ P i ) for some k ≥ 2. We want to show that ¬ k +1 _ i =1 P i = k +1 ^ i =1 ( ¬ P i ) . We compute ¬ k +1 _ i =1 P i = ¬ k _ i =1 P i ∨ P k +1 = ¬ k _ i =1 P i ∧¬ P k +1 (by the usual de Morgan Law) = k ^ i =1 ( ¬ P i ) ∧¬ P k +1 (by inductive hypothesis) = k +1 ^ i =1 ( ¬ P i ) . This concludes the inductive step. Book Problems. Problem 7 Solution. First we show that B ⊆ A c if and only if A ∩ B = ∅ . ( ⇒ ) Suppose that B ⊆ A c , so that x ∈ B ⇒ x / ∈ A . Suppose also, for a contradiction, that A ∩ B 6 = ∅ . Then there exists an element x ∈ A ∩ B . Since x ∈ B it follows that x / ∈ A by our hypothesis. Henceour hypothesis....
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Homework 5 Solutions - HOMEWORK 5 COMMENTS 1 Non-book...

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