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0202-notes - CPSC 121 Lecture 13 February 2 2009 Menu...

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CPSC 121 Lecture 13 February 2, 2009 Menu February 2, 2009 Topics: Proof Techniques (cont’d) Summary of Learning Goals (Epp Chapter 2) (More) Examples Reading: Today: Epp 2.1–2.4 Reminders: In-class (30 minute) Quiz 1 Wednesday, February 4 — one (2-sided) 8.5 × 11 reference sheet allowed Midterm exam Tuesday, February 24 (evening) READ the WebCT Vista course announcements board Proof Techniques (cont’d) We complete our initial discussion of proof techniques by showing an example of proof by contrapositive. Example 1: Theorem: n Z + , if n 2 is even then n is even Proof (by contrapositive): Recall: p q q p Thus, in order to prove x, P ( x ) Q ( x ) we can prove x, Q ( x ) P ( x ) Let E ( x ) : x is even To prove n Z + , E ( n 2 ) E ( n ) we prove n Z + , E ( n ) E ( n 2 ) via a direct proof Example 1: (cont’d) Proof (by contrapositive): Let n Z + be an arbitrary odd integer Therefore, by definition of odd, n = 2 k + 1 for some k Z + Now, n 2 = (2 k + 1) 2 = 4 k 2 + 4 k + 1 = 2(2 k 2 + 2 k ) + 1 Since the sum and product of integers is an integer, we see that m = 2(2 k 2 + 2 k ) is an even integer and n 2 = m + 1 is an odd integer QED Here is a summary of “recipes” for direct proof, proof by contradiction and proof by contrapositive.
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Recipe: Direct Proof 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 P ( x ) , axioms, lemmas and all other theorems as premises) Recipe: Proof by Contradiction 1. Identify the domain of discourse, D 2. Assume the statement to be proven is false (equivalently, assume its negation is true) 3. Then, under this assumption, together with other axioms, lemmas and theorems, derive a contradiction 4. Conclude that the original statement to be proven is true Recipe: Proof by Contrapositive In order to prove x, P ( x ) Q ( x ) prove x, Q ( x ) P ( x ) Summary of Learning Goals (Epp Chapter 2) We have now completed our coverage of the material from Chapter 2 of the textbook. Here is a summary of the learning goals for this second portion of the course.
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