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Unformatted text preview: HOMEWORK 5: COMMENTS 1. Non-book problems (1) Well 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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- Spring '07