1 / 16

Lecture 5 Proof Techniques

Theorem and Informal Techniques. In formal logic we proved arguments that were universally true by their internal structure - arguments that were tautologies (propositional logic) or valid wffs (predicate logic). But, we also need to prove arguments that are not universally true, just within some c

lelia
Download Presentation

Lecture 5 Proof Techniques

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

    1. Lecture 5 Proof Techniques Quiz (15 minutes). Review of Last Class proof of correctness Assignment Rule {Ri} x = e {Ri+1} Conditional Rule From: {Q B} P1 {R}, can derive {Q} Si {R} {Q B} P2 {R} Restriction: Si has the form if condition B then P1 else P2 end if

    2. Theorem and Informal Techniques In formal logic we proved arguments that were universally true by their internal structure - arguments that were tautologies (propositional logic) or valid wffs (predicate logic). But, we also need to prove arguments that are not universally true, just within some context. Subject-specific facts become additional hypotheses for the proof that P ?Q in this particular context There is no formula for constructing proofs. Theorems are often stated and proved in a less formal way.

    3. Background Conjectures (hypothesis, conclusions) is a sentence explaining a general pattern within similar math problems. Like an explanation. Deductive (or logical) Reasoning is the process of demonstrating that if certain statements are accepted as true, then other statements can be shown to follow from them. Inductive Reasoning is the process of observing data, recognizing patterns, and making generalizations from the observations. Counter Example---The example disproves the conjectures

    4. Background (cont.) A single counter example to a conjecture is sufficient to disprove it. N2>4N (N is an integer) Hunting for a counter example and being unsuccessful does not constitute a proof that the conjecture is true. A Famous conjecture-- Goldbach's Conjecture: Every even n > 2 is the sum of two primes.

    5. Background (cont.) Example of a conjecture 21 - 12 =9 83 - 38 = 45 52 - 25 = 27 64 - 46 = 18 Conjuncture--The difference of a two digit number and it's reverse is equal to a multiple of nine.

    6. Deductive Reasoning Moves from the general to the more specific. informally called a "top-down" approach. Try to verify the truth or falsity of a conjecture. Produce a proof P?Q Another option is to find a counterexample that disprove the conjecture.

    7. Inductive Reasoning Moves from specific observations to broader generalizations and theories. Informally, we sometimes call this a "bottom up" approach. Steps: specific observations and measures, detect patterns and regularities formulate some tentative hypotheses that we can explore, end up developing some general conclusions or theories.

    8. Exhaustive Proof Use it when the conjecture is an assertion about a finite collection. The conjecture can be proved true by showing that is true for each member of the collection. Example: For any positive integer less than or equal to 3, the square of the integer is less than or equal to the sum of 10 plus 5 times the integer. n n*n 10+5*n 1 1 15 2 4 20 3 9 25

    9. Direct Proof Assume that the hypothesis P is true and deduce the conclusion Q. A formal proof require a proof sequence leading from P to Q. Example: Prove that the product of two even integers is even. Let x = 2m and y = 2n, where m and n are integers. Then xy = (2m)(2n) = 2(2mn), where 2mn is an integer. Thus xy has the form 2k, where k is an integer, and xy is therefore even.

    10. Contraposition Using the tautology (Q?P)?(P?Q) We can use it to prove (P?Q) By proving the theorem Q ? P we can conclude Q?P. The converse (Q?P) is not equivalent.

    11. When theorems are stated as P if and only if Q we need to prove (P?Q) ^ (Q?P) The truth of one does not imply the truth of the other. For example, if we need to prove that the product xy is odd if and only if both x and y are odd integers, we need to prove .. if x and y are odd, so is xy. if xy is odd, both x and y must be odd. ? (x and y are odd) ? (xy is odd) Contraposition (cont.)

    12. Contraposition (cont.) Proof by case A form of exhaustive proof. It involves identifying all the possible cases consistent with the given information and then proving each case separately. Example: xy odd ? x odd and y odd Use contraposition: (x odd and y odd) ? xy odd De Morgan: x even or y even ? xy even. x even, y odd: x = 2m, y = 2n+1, and then xy=(2m)(2n+1) = (2)(2mn+m), which is even x odd, y even: x = 2m+1, y = 2m, and then xy=(2m+1)(2n) = (2)(2mn+m), which is even x even, y even: x = 2m, y = 2n, and then xy=(2m)(2n)=(2)(2mn) where 2mn is an integer. Thus xy has the form 2k, where k is an integer, and xy is therefore even.

    13. Contradiction We are trying to prove P?Q. We can use the tautology (P^Q?0)?(P?Q) to prove (P?Q). (0 stand for any contradiction) It is sufficient to prove (P^Q?0) We need to assume that the hypothesis and the negation of the conclusion are true and then try to deduce some contradiction from this assumptions. Especially useful if you need to prove something is not true.

    14. Contradiction (cont.) Examples. Sqrt(2) is not a rational number. Prove by contradiction.

    16. Assignment 2- (part a) Exercise 2.1 4, 36, 47, 51

More Related