Pre-Class Learning Goals By the start of class, for each proof strategy below, you CPSC 121: Models of Computation should be able to: Identify the form of statement the strategy can prove. Sketch the structure of a proof that uses the strategy. Strategies for quantifiers: Unit 7: Proof Techniques (for ∀ x ∈ Z . . .) generalizing from the generic particular (WLOG) constructive/non-constructive proofs of existence (for ∃ x ∈ Z . . .) proof by exhaustion (for ∀ x ∈ Z . . .) General strategies (for p → q.) antecedent assumption proof (for p → q.) proof by contrapositive proof by contradiction (for any statement.) proof by cases. (for any statement.) Based on slides by Patrice Belleville and Steve Wolfman Unit 7- Proof Techniques 2 Quiz 7 Feedback: In-Class Learning Goals In general : By the end of this unit, you should be able to: Devise and attempt multiple different, appropriate proof Issues: strategies for a given theorem, including o all those listed in the "pre-class" learning goals o logical equivalences, o propositional rules of inference o rules of inference on quantifiers i.e. be able to apply the strategies listed in the Guide to Proof Strategies reference sheet on the course web site (in Other Handouts) We will do more proof examples in class. For theorems requiring only simple insights beyond strategic choices or for which the insight is given/hinted, additionally prove the theorem. Unit 7- Proof Techniques Unit 7- Proof Techniques 3 4 1
? ? Where We Are in The BIG Questions Unit Outline How can we convince ourselves that an algorithm Techniques for quantifiers . does what it's supposed to do? NOTE: Existential quantifiers. We need to prove its correctness. Epp calls some of these direct proofs How do we determine whether or not one algorithm is Universal quantifiers. and others indirect. better than another one? We’ll avoid using Dealing with multiple quantifiers. these terms Sometimes, we need a proof to convince someone that the number of steps of our algorithm is what we claim it is. Using logical equivalencies : Proof by contrapositive Using Premises Proof by contradiction Additional Examples Unit 7- Proof Techniques 5 Unit 7- Proof Techniques 6 Techniques for quantifiers Existential Statements Suppose the statement has the form : There are two general forms of statements: ∃ x ∈ D, P(x) o Those that start with an existential quantifier. To prove this statement is true, we must o Those that start with a universal quantifier. Find a value of x (a “witness”) for which P(x) holds. We call it a witness proof We use different techniques for them. We’ll study each case in turns. So the proof will look like this: Let x = <some value in D> Verify that the x we chose satisfies the predicate. Example: There is a prime number x such that 3x+2 is not prime. Unit 7- Proof Techniques Unit 7- Proof Techniques 7 8 2
Existential Statements (cont’) Existential Statements (cont’) How do we translate What is the right start of the proof for the statement There is a prime number x such that 3x+2 is not prime There is a prime number x such that 3x+2 is not into predicate logic ? prime ? A. ∀ x ∈ Z + , Prime(x) ∧ ~Prime(3x+2) A. Without loss of generality let x be a positive integer …. B. ∃ x ∈ Z + , Prime(x) ∧ ~Prime(3x+2) B. Without loss of generality let x be a prime …. C. ∀ x ∈ Z + , Prime(x) → ~Prime(3x+2) C. Let x be any non specific prime …… D. ∃ x ∈ Z + , Prime(x) → ~Prime(3x+2) D. Let x be 2 …… E. None of the above. E. None of the above. Unit 7- Proof Techniques 9 Unit 7- Proof Techniques 10 Existential Statements (cont’) Unit Outline So the proof goes as follows: Techniques for direct proofs. Proof: Existential quantifiers. o Let x = Universal quantifiers. o It is prime because its only factors are 1 and o Now 3x+2 = Dealing with multiple quantifiers. and Using logical equivalencies : Proof by contrapositive o Hence 3x+2 is not prime. o QED. Using Premises Proof by contradiction Additional Examples Unit 7- Proof Techniques Unit 7- Proof Techniques 11 12 3
Universal Statements (cont’) Universal Statements Suppose our statement has the form : Terminology: the following statements all mean the ∀ x ∈ D, P(x) same thing: To prove this statement is true, we must Let x be a nonspecific element of D Show that P(x) holds no matter how we choose x. Let x be an unspecified element of D So the proof will look like this: Let x be an arbitrary element of D Let x be a generic element of D Without loss of generality, let x be any element of D Let x be any element of D (or an equivalent expression like those shown on next page) Suppose x is a particular but arbitrarily chosen element of D. Verify that the predicate P holds for this x. o Note: the only assumption we can make about x is the fact that it belongs to D. So we can only use properties common to all elements of D. Unit 7- Proof Techniques 13 Unit 7- Proof Techniques 14 Universal Statements (cont’) Universal Statements (cont’) Example: Every Racket function definition is at least 12 Example 1: Every Racket function definition is at least characters long. 12 characters long. What is the starting phrase of a proof for this statement? The proof goes as follows: Proof: A. Without loss of generality let f be a string of 12 characters …. o Let f be B. Let f be a nonspecific Racket function definition…. o Then f should look like: C. Let f be the following Racket function definition …… D. Let f be a nonspecific Racket function with 12 or more characters …. o Therefore f is at least 12 characters long. E. None of the above. Unit 7- Proof Techniques Unit 7- Proof Techniques 15 16 4
Antecedent Assumption (cont’) Special Case : Antecedent Assumption Why is the line Assume that P(k) is true valid? Suppose the statement has the form: ∀ x ∈ D, P(x) → Q(x) A. Because these are the only cases where Q(k) This is a special case of the previous formula matters. The textbook calls this (and only this) a direct proof. B. Because P(k) is preceded by a universal quantifier. The proof looks like this: C. Because we know that P(k) is true. Proof: o Consider an unspecified element k of D. D. Both (a) and (c) o Assume that P(k) is true. E. Both (b) and (c) o Use this and properties of the element of D to verify that the predicate Q holds for this k. Unit 7- Proof Techniques 17 Unit 7- Proof Techniques 18 Antecedent Assumption (cont’) Antecedent Assumption (cont’) Example 2: The sum of two odd numbers is even. Example: prove that Odd(x) ∃ k ∈ N, x = 2k+1 If ∀ n ∈ N, n ≥ 1024 → 10n ≤ nlog 2 n Even(x) ∃ k ∈ N, x = 2k Proof: the above statement is: WLOG let n be an unspecified natural number. ∀ n ∈ N, ∀ m ∈ N, Odd(n) ᴧ Odd(m) → Even( n+m) Assume that Proof: Then Let n be an arbitrary natural number. Let m be an arbitrary natural number. Assume that n and m are both odd. Then n = 2i+1 for some natural number i, and m = 2j+1 for some natural number j Then m+n = 2i+1 + 2j+1 = 2i + 2j + 2 = 2(i+j+1) Since i+j+1 is a natural number, 2(i+j+1) is even and so is n+m. QED Unit 7- Proof Techniques Unit 7- Proof Techniques 19 20 5
… and for fun … Unit Outline Other interesting proof techniques ☺ Techniques for direct proofs. Proof by intimidation Existential quantifiers. Proof by lack of space (Fermat's favorite!) Universal quantifiers. Proof by authority Proof by never-ending revision Dealing with multiple quantifiers . Using logical equivalencies : Proof by contrapositive For the full list, see: http://school.maths.uwa.edu.au/~berwin/humour/invalid.proo Using Premises fs.html Proof by contradiction Additional Examples Unit 7- Proof Techniques 21 Unit 7- Proof Techniques 22 Multiple Quantifiers Multiple Quantifiers: Example Theorem: i Z + , n Z + , n i 60n < n 2 How do we deal with theorems that involve multiple quantifiers? We can think of it as a statement of the form Start the proof from the outermost quantifier. i Z + , P(i) , Work our way inwards. where P(i) n Z + , n i 60n < n Example: Suppose we wan to prove: So, how do we pick i An algorithm whose run time is t(n) = 60n is generally faster than an algorithm whose time is n 2 , i.e. we want to show that A. Let i be any specific integer. as n increases, 60n < n 2 B. Without loss of generality, let i be any arbitrary positive integer The statement in predicate logic is: i Z + , n Z + , n i 60n < n 2 C. Let i = (a specific value) D. None of the above Unit 7- Proof Techniques Unit 7- Proof Techniques 23 24 6
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