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Transcript

Chapt 2.1 Proof Techniques Reading this week: 2.1 Proofs, Recursion and Analysis of Algorithms Motivations 1 Formal logic allows to perform detailed, direct proofs of the validity of arguments. However, proofs in formal logic can be rather long and complex to perform. Often a more informal proof is sufficient. In many situations we are not interested in the validity of an argument but if a conjecture is correct in a given interpretation. We have to be able to proof not only if something is correct but also if it is wrong. To be able to do this and to facilitate the proof of a large range of problems we need a variety of different proof techniques. Proof Techniques 1 Informal Proofs 2 In general we want to prove a conjecture P Q in a specific context. Once a conjecture is proven it becomes a theorem. Conjectures are often created based on inductive reasoning. Inductive reasoning describes the process of drawing conclusions based on a number of experiences. To prove a conjecture we have to apply deductive reasoning where the truth or falsity is verified by evaluating or transforming the conjecture. Proof Techniques Informal Proofs 3 Proofs are often presented in a narrative form, taking advantage of the properties of the specific context to allow shorter proofs. – Note: Informal proofs have to contain enough detail to be reproduceable and to be translated into a formal proof by introducing the rules specific to the context. – In informal proofs conjectures like ( x)(P(x) Q(x)) are mostly formulated as P(x) Q(x) where the quantifier is maintained only implicitly. In formal logic this would be achieved by applying universal instantiation and then later universal generalization. Proof Techniques 2 Proof Techniques 4 Informal proof methods : Disproof by counterexample Proof by exhaustion Direct proof Proof by contraposition Proof by contradiction Proof Techniques Terms 5 A few terms to remember: Axioms: Statements that are assumed true. • Example: Given two distinct points, there is exactly one line that contains them. Definitions: Used to create new concepts in terms of existing ones. Theorem: A proposition that has been proved to be true. • Two special kinds of theorems: Lemma and Corollary. • Lemma: A theorem that is usually not too interesting in its own right but is useful in proving another theorem. • Corollary: A theorem that follows quickly from another theorem. Proof Techniques 3 Proof by Counterexample 6 Inductive Reasoning: Drawing a conclusion from a hypothesis based on experience. Hence the more cases you find where P follows from Q, the more confident you are about the conjecture P Q. Deductive reasoning is applied to the same conjecture to ensure that it is indeed valid. Deductive reasoning can look for a counterexample that disproves the conjecture, i.e. a case when P is true but Q is false. Proof Techniques Counterexample 7 To prove that a conjecture is false it is sufficient to find a single example which contradicts the conjecture. Example: Prove or disprove: For any positive integer n, n2 10 + 5n. n n2 10+5n n2 < 10+5n 0 0 10 yes 1 1 15 yes 2 4 20 yes 3 9 25 yes 4 16 30 yes 5 25 35 yes 6 36 40 yes 7 49 45 False Proof Techniques 4 Counterexample 8 More examples of counterexample: All animals living in the ocean are fish. Every integer less than 10 is bigger than 5. There is no general way to find a specific counterexample. Moreover, there is no mechanical way to determine if a conjecture is true or false. Counter example is not trivial for all cases, so we have to use other proof methods. Proof Techniques Exhaustive Proof 9 If dealing with a finite domain in which the proof is to be shown to be valid, then using the exhaustive proof technique, one can go over all the possible cases for each member of the finite domain. Final result of this exercise: you prove or disprove the theorem but you could be definitely exhausted. This proof technique can only be applied if the set of cases is finite. One example of an exhaustive proof technique are truth tables in propositional logic. Proof Techniques 5 Example: Exhaustive Proof 10 Example: For any positive integer less than or equal to 5, the square of the integer is less than or equal to the sum of 10 plus 5 times the integer. n n2 10+5n n2 < 10+5n 0 0 10 yes 1 1 15 yes 2 4 20 yes 3 9 25 yes 4 16 30 yes 5 25 35 yes Proof Techniques Example: Exhaustive Proof 11 Example : Every even integer between 4 and 6 is the product of exactly 2 prime numbers. Number of cases: 4=2*2 6=2*3 Proof Techniques 6 Direct Proof 12 Direct Proof: Used when exhaustive proof doesn’t work. Using the rules of propositional and predicate logic, prove P Q. Hence, assume the hypothesis P and prove Q. Hence, a formal proof would consist of a proof sequence to go from P to Q. Proof Techniques Direct Proof Example 13 Example: If an integer is divisible by 6, then twice that integer is divisible by 4. Proof: Since x is divisible by 6, x can be written as x = 6k where k is an integer. We have 2x = 2 * (6k) = 12k = 4 *(3k). Since 3k is an integer, 2x is divisible by 4. Proof Techniques 7 Direct Proof Example 14 Consider the conjecture : x is an even integer Λ y is an even integer the product xy is an even integer. OR: (x) (y)(x is an even integer Λ y is an even integer xy is an even integer) Formal Proof: A complete formal proof sequence might look like the following: 1. x is an even integer Λ y is an even integer hyp 2. (x)[x is even integer (k)(k is an integer Λ x = 2k)] number fact (definition of even integer) 3. x is even integer (k)(k is an integer Λ x = 2k) 2,ui 4. y is even integer (k)(k is an integer Λ y = 2k) 2,ui 5. x is an even integer 1,sim 6. (k)(k is an integer Λ x = 2k) 3,5,mp 7. m is an integer Λ x = 2m 6, ei 8. y is an even integer 1,sim Proof Techniques 15 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. Direct Proof Example (contd.) (k)(k is an integer Λ y = 2k) 4,8,mp n is an integer and y = 2n 9, ei x = 2m 7,sim y = 2n 10, sim xy = (2m)(2n) 11, 12, substitution of equals xy = 2(2mn) 13, multiplication fact m is an integer 7,sim n is an integer 10, sim 2mn is an integer 15, 16, number fact xy = 2(2mn) Λ 2mn is an integer 14, 17, con (k)(k is an integer Λ xy = 2k) 18,eg (x)((k)(k is an integer Λ x = 2k) x is even integer) number fact (definition of even integer) (k)(k an integer Λ xy = 2k) xy is even integer 20, ui xy is an even integer 19, 21, mp Proof Techniques 8 Direct Proof Example (cont’d.) 16 Direct Proof: The above lengthy proof can be done by using narrative and proved using a shorter informal direct proof. Proof: Let x = 2m and y = 2n, where m and n are integers. Then xy = (2m)(2n) = 2 (2mn) = 2k where k= 2mn and is an integer. Thus xy has the form 2k. Based on the definition of even number, xy is therefore even. Proof Techniques Proof by Contraposition 17 Use the variants of P Q to prove the conjecture. From the inference rules of propositional logic, we know a rule called contraposition (termed “cont” in short). It states that (Q P) (P Q) (a Tautology). Q P is the contrapositive of P Q. Hence, proving the contrapositive implies proving the conjecture. The contrapositive is different from the converse of a conjecture. The converse Q P is not tautologically equivalent to the conjecture P Q and proving the converse does not prove the conjecture. Proof Techniques 9 Proof by Contraposition 18 Example: Prove that if the square of an integer is odd, then the integer must be odd Hence, prove n2 odd n odd To do a proof by contraposition prove n even n2 even If n is even, then it can be written as 2m where m is an integer, i.e. even/odd. Then, n2 = 4m2 which is always even no matter what m2 is because of the factor 4. Proof Techniques Example: Proof by Contraposition 19 Example: If x is positive, so is x + 1. Example: If there are more than n assignment sheets given out to n students, then some student gets more than one assignment sheet. Proof Techniques 10 Indirect Proof: Proof by Contradiction 20 Proof by contradiction uses again a transformation of the original conjecture as P Λ Q. Here the argument P Λ Q 0 is proven. Derived from the definition of contradiction which says P Λ Q 0 we use the fact that (P Λ Q 0) (P Q) is a tautology. In a proof of contradiction, one assumes that the hypothesis and the negation of the conclusion are true and then to deduce some contradiction from these assumptions. Proof Techniques Proof by Contradiction 21 Example 1: Prove that “If a number added to itself gives itself, then the number is 0.” The hypothesis (P) is x + x = x and the conclusion (Q) is x = 0. Hence, the hypotheses for the proof by contradiction are: x + x = x and x 0 (the negation of the conclusion) Then 2x = x and x 0, hence dividing both sides by x, the result is 2 = 1, which is a contradiction. Hence, (x + x = x) (x = 0). Proof Techniques 11 Proof by Contradiction 22 Example 2: Prove “For all real numbers x and y, if x + y 2, then either x 1 or y 1.” Proof Techniques Proof by Contradiction 23 Example 3: The sum of even integers is even. Proof: Let x = 2m, y = 2n for integers m and n P: (x is even) Λ (y is even) Q: (x + y) is even. Q’ assumes that x + y is odd. From original P we have x + y = 2m + 2n , From Q’, we have x + y = 2k + 1 So we have x + y = 2m + 2n = 2k + 1 for some integer k. Hence, 2*(m + k - n) = 1, where m + n - k is some integer. This is a contradiction since 1 is not even. Proof Techniques 12 Summarizing Proof techniques 24 Proof Technique Approach to prove P Q Remarks Exhaustive Proof Demonstrate P Q for all cases May only be used for finite number of cases Direct Proof Assume P, deduce Q The standard approachusually the thing to try Proof by Contraposition Assume Q, derive P Use this Q if as a hypothesis seems to give more ammunition then P would Proof by Contradiction Assume P Λ Q , deduce a contradiction Use this when Q says something is not true Serendipity Not really a proof technique Fun to know Proof Techniques Exercise: Which Proof technique is used? 25 Example 1: Conjecture: There are more odd natural numbers smaller than 10 that there are even ones. Proof: There are 5 odd natural numbers smaller than 10 (1, 3, 5, 7, and 9). There are 4 even ones (2, 4, 6, and 8). Therefore there are more odd natural numbers smaller than 10 than there are even ones. Proof Technique: Proof Techniques 13 Exercise: Which Proof technique is used? 26 Example 2: Conjecture: The sum of an even and an odd integers is odd. Proof: Let n = 2* k and m = 2 * h + 1 be an even and an odd integer, respectively. (k and h are integers). Then their sum is m+n = 2*k +2 *h + 1 = 2 (k +h)+1 and thus odd. Proof Technique: Proof Techniques Exercise: Which Proof technique is used? 27 Example 3: Conjecture: If the product of two integers is even, then at least one of the integers is even. Proof: Assume both integers are odd, then their product is m = (2 *k+1) (2*h+1) = 4kh +2k + 2h + 1 = 2 (2kh +k +h) + 1 and thus odd. Proof Technique: Proof Techniques 14 Exercise: Which Proof technique is used? 28 Example 4: Conjecture: The sum of two odd integers is even. Proof: Suppose the sum m is odd. Then there is an integer k such that m = 2 * k+1. Since m is the sum of two integers, it can be written as 2 * k+1 = x + y and therefore at least one of the two integers has to be even. Proof Technique: Proof Techniques 15