hw2math109

hw2math109 - MATH 109 Due: 4/11/11 Assignment 2 #4.2 Proof....

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MATH 109 Assignment 2 Due: 4/11/11 #4.2 Proof. We are given that n is odd, and we suppose for a contradiction that n 2 is not odd. By definition, this implies that 2 6 | n and 2 | n 2 . Let x Z be the integer such that 2 x = n 2 . Observe that ( n + 2) 2 = n 2 + 4 n + 4 = 2 x + 4 n + 4 = 2( x + 2 n + 2) so ( n + 2) 2 is even. By Part I Problem #7, we have that n + 2 must be even so let y Z be the integer such that 2 y = n + 2. Since n = ( n + 2) - 2 = 2 y - 2 = 2( y - 1) , we get a contradiction to the fact that n is odd. Hence, n 2 must be odd. #4.7 Proof. By Axiom 3.1.2 (ii), we can add the inequalities -| a | ≤ a ≤ | a | and -| b | ≤ b ≤ | b | to obtain - ( | a | + | b | ) a + b ≤ | a | + | b | , and hence we conclude that | a + b | ≤ | a | + | b | . A necessary and sufficient condition for equality is that either (i) ( a = 0 or b = 0) or (ii) a = cb for some positive c R .
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hw2math109 - MATH 109 Due: 4/11/11 Assignment 2 #4.2 Proof....

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