W2009 - Math 135 1. Midterm Solutions Winter 2009 a) Put...

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Math 135 Midterm Solutions Winter 2009 1. a) Put the following statement into logical/mathematical notation (using quantifiers). ”There is no largest integer.” State the universe of discourse. Not ( x y, x y ) or x y, y > x Universe of discourse is the integers. b) If the universe of discourse is the integers, is the statement ” x y, x 3 + y = 0” true or false? Explain. True. Take any integer x , then x 3 is an integer and setting y = - x 3 gives x 3 + y = 0. 2. Let A be the statement ”If n is an odd integer, then n ( n - 1) is even”. a) Is A true or false? Explain. True. If n is odd, then ( n - 1) is even and the product of an even integer with and odd integer is even. OR Since n is odd, n can be expressed in the form 2 k + 1 for some k Z . Thus, n ( n - 1) = (2 k + 1)(2 k ) = 2(2 k 2 + k ), an even integer. Therefore, statement A is true. b) State the converse of A . If n ( n - 1) is even, then n is an odd integer. c) Is the converse of A true or false? Explain. False. If n is even, n ( n - 1) is even also. OR Counterexample: 4(3) = 12 which is even, but 4 is not odd. d) State the contrapositive of A . If n ( n - 1) is odd, then n is an even integer.
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3. Prove by induction that n X i =1 i · ( i !) = ( n + 1)! - 1. Base Case, n = 1 1 X i =1 i · ( i !) = 1 · 1! = 1, and (1 + 1)! - 1 = 2! - 1 = 2 - 1 = 1 The Base Case is true.
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This note was uploaded on 11/25/2010 for the course MATH 135 taught by Professor Andrewchilds during the Winter '08 term at Waterloo.

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W2009 - Math 135 1. Midterm Solutions Winter 2009 a) Put...

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