0130-notes - CPSC 121 Lecture 12 January 30, 2009 Menu...

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CPSC 121 Lecture 12 January 30, 2009 Menu January 30, 2009 Topics: Proof Techniques — Direct Proofs — Indirect Proofs Reading: Today: Epp 2.2–2.4 Reminders: Assignment 1 due TODAY (by 17:00) On-line Quiz 6 deadline 9:00pm February 1 In-class Quiz 1 Wednesday, February 4 Midterm exam Tuesday, February 24 (evening) READ the WebCT Vista course announcements board Here’s the example we finished off with last time. Example 1: Prove that there exist irrational numbers x and y such that x y is rational Let R ( x ) : x is a rational number (i.e., x Q ) Prove: x R , y R , R ( x ) R ( y ) R ( x y ) Proof: We know 2 is irrational (we prove this later) Consider 2 2 . If it is rational, then we’re done Otherwise, let x = 2 2 and y = 2 so that x y = 2 Therefore, either the pair x = 2 and y = 2 or the pair x = 2 2 and y = 2 satisfy R ( x ) R ( y ) R ( x y ) QED Example 1 invokes the number 2 2 . We do not know whether this number is rational or irrational. Either way, we are able to use 2 2 to prove the existence of irrational numbers x and y such that x y is rational. Note: The fact that ( 2 2 ) 2 = ( 2) 2 2 = ( 2) 2 = 2 is used as part of the proof. Proof Techniques Direct Proofs Most of the deductive arguments we have seen so far have been are examples of Direct Proofs . Recipe for a Direct Proof While each theorem differs in details, there are common forms Theorems often take the form
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x,P ( x ) Q ( x ) To prove this type of theorem, we must show that P ( x ) Q ( x ) is always true P ( x ) Q ( x ) is false only if, for some x , P ( x ) is true and Q ( x ) is false Thus, we assume P ( x ) to be true and show Q ( x ) must be true when P ( x ) is true ASIDE: We don’t care if P ( x ) is false since P ( x ) Q ( x ) then is true (vacuously) Direct Proofs To summarize, the steps in a direct proof are: 1. Identify the domain of discourse, D 2. Assume that x is an arbitrary element of D for which P ( x ) is true 3. Then, under this assumption, show that Q ( x ) is true (using a valid argument with
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This note was uploaded on 01/09/2010 for the course CPSC 121 taught by Professor Belleville during the Spring '08 term at The University of British Columbia.

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0130-notes - CPSC 121 Lecture 12 January 30, 2009 Menu...

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