This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: HOMEWORK 5: COMMENTS 1. Nonbook 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....
View
Full
Document
 Spring '07
 COURTNEY
 Division

Click to edit the document details