1. Lecture 10 Methods of Proof CSCI – 1900 Mathematics for Computer Science Fall 2014 Bill Pine

2. CSCI 1900 Lecture 10 - 2 Lecture Introduction Reading –Rosen - Section 1.7, 1.8 The Nature of Proofs Components of a Proof Rule of Inference and Tautology Proving equivalences modus ponens Indirect Method Proof by Contradiction

3. CSCI 1900 Lecture 10 - 3 Past Experience Until now we’ve used the following methods to write proofs: –Direct proofs with generic elements, definitions, and given facts –Proof by enumeration of cases such as when we used truth tables

4. CSCI 1900 Lecture 10 - 4 General Outline of a Proof p  q denotes "q logically follows from p“ Implication may take the form (p 1  p 2  p 3  …  p n )  q q logically follows from p 1, p 2, p 3, …, p n

5. CSCI 1900 Lecture 10 - 5 General Outline (cont) The process is generally written as: p 1 p 2 p 3 : : p n  q

6. CSCI 1900 Lecture 10 - 6 Components of a Proof The p i 's are called hypotheses or premises q is called the conclusion Proof shows that if all of the p i 's are true, then q has to be true If result is a tautology, then the implication p  q represents a universally correct method of reasoning and is called a rule of inference

7. CSCI 1900 Lecture 10 - 7 Example of a Tautology based Proof If p implies q and q implies r, then p implies r p  q q  r  p  r By replacing the bar under q  r with the “  ”, the proof above becomes ((p  q)  (q  r))  (p  r) The next slide shows that this is a tautology and therefore is universally valid.

8. CSCI 1900 Lecture 10 - 8 Tautology Example (cont) pqr p  qq  r(p  q)  (q  r) p  r((p  q)  (q  r))  (p  r) TTTTTTTT TTFTFFFT TFTFTFTT TFFFTFFT FTTTTTTT FTFTFFTT FFTTTTTT FFFTTTTT

9. CSCI 1900 Lecture 10 - 9 Equivalences Some mathematical theorems are equivalences, i.e., p  q. –If and only if –Necessary and sufficient The proof of such a theorem requires two proofs: –Must prove p  q, and –Must prove q  p

10. CSCI 1900 Lecture 10 - 10 Rule of Inference: modus ponens modus ponens – the method of asserting p p  q  q Example: –p: It is snowing today –p  q: If it is snowing, there is no school –q: There is no school today Supported by the tautology (p  (p  q))  q

11. CSCI 1900 Lecture 10 - 11 Show modus ponens = Rule of Inference pq (p  q)p  (p  q)(p  (p  q))  q TTTTT TFFFT FTTFT FFTFT

12. CSCI 1900 Lecture 10 - 12 Summary of Arguments An argument is a sequence of statements –All statement except the last are premises –The final statement is the conclusion –Normally place the symbol  in front of the conclusion To say that an argument is valid means only that its form is valid –If the premises are all true the conclusion is true

13. CSCI 1900 Lecture 10 - 13 Summary of Arguments (cont) It is possible for a valid argument to lead to a false conclusion It is possible for an invalid argument to lead to a true conclusion Carefully distinguish between validity of the argument and truth of the conclusion A valid argument is one whose form, when supplied with true premises cannot have a false conclusion

14. CSCI 1900 Lecture 10 - 14 Invalid Conclusions from Invalid Premises Just because the format of the argument is valid does not mean that the conclusion is true. A premise may be false. For example: Star Trek is real If Star Trek is real we can travel faster than light  We can travel faster than light Argument is valid since it is in modus ponens form Conclusion is false because a premise is false

15. CSCI 1900 Lecture 10 - 15 Invalid Conclusion from Wrong Form Sometimes, an argument that looks like modus ponens is actually not in the correct form. For example: If I am watching Star Trek, then I am happy I am happy  I am watching Star Trek Argument is not valid since its form is: p  q q  p Which is not modus ponens

16. CSCI 1900 Lecture 10 - 16 Invalid Argument (continued) Truth table shows that this is not a tautology: pq (p  q)(p  q)  q((p  q)  q)  p TTTTT TFFFT FTTTF FFTFT

17. CSCI 1900 Lecture 10 - 17 Rule of Inference: Indirect Method Another method of proof is to use the tautology: (p  q)  (~q  ~p) The form of the proof is (modus ponens): ~q ~q  ~p  ~p

18. CSCI 1900 Lecture 10 - 18 Indirect Method Example p: My e-mail address is available on a web site q: I am getting spam p  q: If my e-mail address is available on a web site, then I am getting spam ~q  ~p: If I am not getting spam, then my e-mail address must not be available on a web site This proof says that if I am not getting spam, then my e-mail address is not on a web site

19. CSCI 1900 Lecture 10 - 19 Another Indirect Method Example Prove that if the square of an integer is odd, then the integer is odd too. p: n 2 is odd q: n is odd ~q  ~p: If n is even, then n 2 is even. If n is even, then there exists an integer m for which n = 2×m. n 2 therefore would equal (2×m) 2 = 4×m 2 which must be even.

20. CSCI 1900 Lecture 10 - 20 Rule of Inference:modus tollens Another method of proof is to use the tautology (p  q)  (~q)  (~p) The form of the proof is: p  q ~q  ~p Also known as proof by contradiction

21. CSCI 1900 Lecture 10 - 21 Proof by Contradiction (cont) pq (p  q) ~q (p  q)  ~q ~p (p  q)  (~q)  (~p) TTTFFFT TFFTFFT FTTFFTT FFTTTTT

22. CSCI 1900 Lecture 10 - 22 Proof by Contradiction (cont) The best application for this is where you cannot possibly go through a large (potentially infinite) number of cases to prove that every one is true

23. CSCI 1900 Lecture 10 - 23 Key Concepts Summary The Nature of Proofs Components of a Proof Rule of Inference and Tautology Proving equivalences modus ponens Indirect Method Proof by Contradiction