contrapositivekey - McCombs Math 381 Contrapositive and...

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McCombs Math 381 Contrapositive and Contradiction Proof by Contrapositive: To prove “if A, then B”: Assume not B, and then show not A. Proof by Contradiction: To prove “if A, then B”: Assume A and not B, and then show you get a contradiction . Prove each of the following by the contrapositive method. 1. If x and y are two integers whose product is even, then at least one of the two must be even. : A x y ! = even : A ¬ x y ! = odd : B x = even or y = even : B ¬ x = odd and y = odd We need to show that B A ¬ ! ¬ . x = odd and y = odd means that 2 1 x k = + and 2 1 y m = + , for some integers k and m . Thus, ( )( ) ( ) 2 1 2 1 4 2 2 1 2 2 1 x y k m km k m km k m ! = + + = + + + = + + + . Therefore, x y ! = odd, by the definition of an odd integer. Hence, we have shown that B A ¬ ! ¬ . 2. If x and y are two integers whose product is odd, then both must be odd. : A x y ! = odd : A ¬ x y ! = even : B x = odd and y = odd : B ¬ x = even or y = even We need to show that B A ¬ ! ¬ . x = even or y = even means that 2 x k = or 2 y m = , for some integers k and m . If 2 x k = , then ( ) ( ) 2 2 x y k y ky ! = = . Thus, x y ! = even, by the definition of an even integer. Similarly, if 2 y m = , then ( ) ( ) 2 2 x y x m xm ! = = . Thus, x y ! = even, by the definition of an even integer.
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contrapositivekey - McCombs Math 381 Contrapositive and...

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