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Unformatted text preview: Methods of Proof One way of proving things is by induction. • That’s coming next. What if you can’t use induction? Typically you’re trying to prove a statement like “Given X , prove (or show that) Y ”. This means you have to prove X ⇒ Y In the proof, you’re allowed to assume X , and then show that Y is true, using X . • A special case: if there is no X , you just have to prove Y or true ⇒ Y . Alternatively, you can do a proof by contradiction : As- sume that Y is false, and show that X is false. • This amounts to proving ¬ Y ⇒ ¬ X 1 Example Theorem n is odd iff (in and only if) n 2 is odd, for n ∈ Z . Proof: We have to show 1. n odd ⇒ n 2 odd 2. n 2 odd ⇒ n odd For (1), if n is odd, it is of the form 2 k + 1. Hence, n 2 = 4 k 2 + 4 k + 1 = 2(2 k 2 + 2 k ) + 1 Thus, n 2 is odd. For (2), we proceed by contradiction. Suppose n 2 is odd and n is even. Then n = 2 k for some k , and n 2 = 4 k 2 . Thus, n 2 is even. This is a contradiction. Thus, n must be odd. 2 A Proof By Contradiction Theorem: √ 2 is irrational. Proof: By contradiction. Suppose √ 2 is rational. Then √ 2 = a/b for some a,b ∈ N + . We can assume that a/b is in lowest terms. • Therefore, a and b can’t both be even. Squaring both sides, we get 2 = a 2 /b 2 Thus, a 2 = 2 b 2 , so a 2 is even. This means that a must be even. Suppose a = 2 c . Then a 2 = 4 c 2 . Thus, 4 c 2 = 2 b 2 , so b 2 = 2 c 2 . This means that b 2 is even, and hence so is b . Contradiction! Thus, √ 2 must be irrational. 3 Induction This is perhaps the most important technique we’ll learn for proving things. Idea: To prove that a statement is true for all natural numbers, show that it is true for 1 ( base case or basis step ) and show that if it is true for n , it is also true for n + 1 ( inductive step ). • The base case does not have to be 1; it could be 0, 2, 3, ... • If the base case is k , then you are proving the state- ment for all n ≥ k . It is sometimes quite difficult to formulate the statement to prove. IN THIS COURSE, I WILL BE VERY FUSSY ABOUT THE FORMULATION OF THE STATEMENT TO PROVE. YOU MUST STATE IT VERY CLEARLY. I WILL ALSO BE PICKY ABOUT THE FORM OF THE INDUC- TIVE PROOF. 4 Writing Up a Proof by Induction 1. State the hypothesis very clearly: • Let P ( n ) be the (English) statement ...[some state- ment involving n ] 2. The basis step • P ( k ) holds because ...[where k is the base case, usually 0 or 1] 3. Inductive step • Assume P ( n ). We prove P ( n +1) holds as follows ...Thus, P ( n ) ⇒ P ( n + 1). 4. Conclusion • Thus, we have shown by induction that P ( n ) holds for all n ≥ k (where k was what you used for your basis step). [It’s not necessary to always write the conclusion explicitly.] 5 A Simple Example Theorem: For all positive integers n , n summationdisplay k =1 k = n ( n + 1) 2 ....
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This note was uploaded on 10/09/2008 for the course COM S 280 taught by Professor Kleinberg during the Spring '05 term at Cornell.
- Spring '05