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: MATH 240; EXAM # 1, 100 points, October 17, 2005 (R.A.Brualdi) TOTAL SCORE (10 problems; 100 points possible): Name: These R. Solutions TA: Luanlei Zhao (circle time) Mon 9:55 Mon 11:00 Wed 9:55 Wed. 11:00 1. [10 points] Using the method of proof by contradiction , give a proof of the following proposition: If n is a positive integer, then not both n and n + 2 are perfect squares. Suppose n is a positive integer, and that both n and n + 2 are perfect squares. Then n = k 2 and n + 2 = l 2 where l > k . Since n + 2 is 2 more than n , we have k 2 + 2 = l 2 and so 2 = l 2 k 2 = ( l + k )( l k ) . Since 2 is a prime, l + k = 2 and l k = 1. Adding these two equations we get 2 l = 3 or l = 3 / 2, contradicting that l is an in integer. This contradiction shows that not both n and n + 2 are perfect squares. 2. [8 points] Let P ( x ) and Q ( x ) be predicates where the universe of discourse for x is some set U . Let A = { x : P ( x ) is true } and let B = { x : Q ( x ) is true } be the truth sets of P ( x ) and Q ( x ), respectively. Circle all the predicates below that have truth set equal to A B ?...
View
Full
Document
This note was uploaded on 08/11/2008 for the course MATH 240 taught by Professor Miller during the Fall '08 term at Wisconsin.
 Fall '08
 Miller
 Math

Click to edit the document details