Class Note for EECS 210 at KU
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This 3 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Kansas taught by a professor in Fall. Since its upload, it has received 14 views.
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Date Created: 02/06/15
EECS 210 Spring 2009 Section 17 Proof Methods and Strategy Proof Methods 0 Proof by Cases 7 Exhaustive Proofs 7 Without Loss of Generality 0 Existence Proof 7 Constructive 7 NonConstructive o Uniqueness Proof Proof Strategies 0 Forward and Backward Reasoning o Adapting Existing Proofs 0 Looking for Counterexarnples 1 Proof by Cases ldea p1Vp2VmVpnH4llt gtp1H4Ap2eqNAanql 11 Exhaustive Proofs Example 1 Prove that n l2 2 3 if n is a positive integer With n S 2 ldea Try for n 172 Example 2 Prove that the only consecutive positive integers not exceeding 100 that are perfect powers are 8 and 9 Note that an integer is a perfect power if it equals n Where a is an integer greater than 1 ldea List all powers til 100 and check Note Easier With computer 12 Non Exhaustive Example 3 Prove that if n is an integer7 then n2 2 n ldea Separate cases for n 07 ii n S 71 and iii n 2 1 Example 4 Let n be an integer Prove that 9n2 3n 7 2 is even ldea Factor the polynomial7 and separate cases for parity of one factor 121 Without Loss of Generality Example 5 Exercise 5 Prove the triangle inequality which states that if z and y are real numbers7 then 2 lz Note that absolute value lTl is equal to T if T 2 0 and is equal to 77 is T lt 0 ldea Just consider cases for z and y to be negative or nonnegative Example 6 PRove that ldea Just consider cases for z and y to be negative or nonnegative 2 Existence Proofs How to prove that some object With particular properties exists 21 Constructive Example 7 Show that there is a positive integer that can be Written as the sum of cubes of positive integers in two different ways ldea Search and ndit71729 103 93 123 13 2 2 NonConstructive Example 8 Assuming that is irrational7 show that there exist irrational numbers I and y such that my is rational Idea Consider cases for z y and then for z and y
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