37 Proof Techniques

37 Proof Techniques - Handout#37 Feb 15 2008 CS103A Robert...

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Handout #37 CS103A Feb. 15, 2008 Robert Plummer Proof Techniques In this handout we will discuss various proof techniques for mathematical proofs. Quantifiers 1: Construction Proofs We will first consider existential conditionals. They have the form: There is an object with a certain property such that something is true. The first step in proving these statements is recognizing the object, property, and the something that is true. Then you proceed by using the construction method. The basic idea is just to construct the object, which makes the statement true. In addition, you must show that the object has the "certain property" and the "something is true". But, we only need one object to prove the statement true. Sometimes you construct the object by trial and error, or you might develop an algorithm to produce the desired object. Prove: There exists an integer n that can be written in two ways as a sum of two prime numbers. object: integer n property: none really except that n is an integer something is true: n can be written two ways as a sum of two prime numbers. By trial and error, n = 10: 5 + 5 = 10 and 7 + 3 = 10. Therefore, there exists an integer n that can be written in two ways as a sum of two prime numbers. Prove: There exists a unique prime number of the form n 2 – 1 where n is an integer 2. Existence Proof: When n = 2, n 2 – 1= 3 which is a prime. Uniqueness Proof: Suppose m is another integer that satisfies this condition and m > 2. Factor m 2 – 1 to get (m – 1)(m + 1). We know that m > 2 and so m – 1 > 1 and m + 1 > 1. Therefore m 2 – 1 cannot be prime since neither of these factors can equal 1. This is a contradiction. Quantifiers II: Counterexample and Choose Proofs with universal quantifiers often have this form: For all x in the set S, if A(x) then B(x). The first step in proving such a conditional is to rewrite the statement to be proven to get it into this form. Then, we can proceed in different ways. One is to find a counterexample thereby proving the statement is false (it doesn't work for all).
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2 Prove: All primes are odd. New statement: For all integers k, if k is prime, then k is odd. 2 is a prime number and is not odd. Thus, by counterexample all primes are not odd. Prove: For any real numbers x and y, floor(x-y) = floor(x) – floor(y). Another method of proof is called the choose method, which is based on the following idea: To show that every element in a set satisfies a particular property, suppose an element x is a particular but arbitrarily chosen element of the set, and show that x satisfies that property. The point of having x be arbitrarily chosen (or generic) is to make the proof general, i.e., you are making no special assumptions about x that are not also true of all the other elements of S. You probably recognize this from FOL as universal introduction. So, whatever you prove about x, you can prove about any other element in the set.
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This note was uploaded on 10/01/2011 for the course CS 103A taught by Professor Plummer,r during the Winter '07 term at Stanford.

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37 Proof Techniques - Handout#37 Feb 15 2008 CS103A Robert...

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