methodsofproofkey - McCombs Math 381 Methods of Proof...

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McCombs Math 381 Methods of Proof 1 Direct Proof: To prove p q : Assume p is true , then show this leads to q being true. Proof by Contrapositive: To prove p q : Assume q is false , then show this leads to p being false. That is, show that ¬ q → ¬ p . Proof by Contradiction: To prove p q : Assume p is true and q is false , then show this leads to a contradiction of the original assumption, or of some other already established result. That is, show that p ∧¬ q ( ) contradiction . This method demonstrates ¬ p q ( ) F . Proof by Cases: To prove p q , where p = p 1 p 2 p 3 ∨⋅⋅⋅∨ p n : S h o w p 1 q , p 2 q , p 3 q , ⋅⋅⋅ p n q . That is, show each case comprising p leads to q . Note: This method requires a finite number of cases. Constructive Existence To prove xP x ( ) : Proof: Find (i.e. “construct”) a specific element of the domain s e t f o r w h i c h P is true. Non-Constructive Existence To prove xP x ( ) : Proof: Demonstrate that P is true without specifically finding an element of the domain set.
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McCombs Math 381 Methods of Proof 2 Prove each of the following by the direct proof . 1. Every odd integer is the difference of two squares. Given integer n . : A n is odd : B 22 np q = for some pair of integers , p q Prove : AB . Consider the odd integer n . Since n is odd, 21 nk = for some integer k . Note that () 2 2 12 1 2 1 kk k k k k −− = −+ −= − . Since k is an integer, 1 k is an integer. Thus, n is the difference of two squares. 2. The sum of an even integer and an odd integer is odd. Given integers n and m . If n is even, and m is odd, then nm + is odd. : A n is even, and m is odd : B + is odd. Prove : A B . Assume integer n is even, and integer m is odd. This means 2 = for some integer k , and mp =+ for some integer p . Thus ( ) 221 2 1 nm k p k p + =++ = ++ . Since k and p are integers, kp + is an integer. Therefore, + is odd. Prove each of the following by the contrapositive method. 1. Given integers x and y . If x y 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 To prove , we need to show that B A ¬ →¬ .
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This note was uploaded on 06/16/2011 for the course MATH 381 taught by Professor Na during the Summer '11 term at University of North Carolina School of the Arts.

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methodsofproofkey - McCombs Math 381 Methods of Proof...

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