Lecture 10 Methods of Proof

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

2. Lecture Introduction • Reading • Kolman - Section 2.3 • The Nature of Proofs • Components of a Proof • Rule of Inference and Tautology • Proving equivalences • modus ponens • Indirect Method • Proof by Contradiction CSCI 1900

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 CSCI 1900

4. General Outline of a Proof • pq denotes "q logically follows from p“ • Implication may take the form (p1 p2 p3 … pn) q • q logically follows from p1, p2, p3, …,pn CSCI 1900

5. General Outline (cont) The process is generally written as: p1 p2 p3 : : pn  q CSCI 1900

6. Components of a Proof • The pi's are called hypotheses or premises • q is called the conclusion • Proof shows that if all of the pi'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 CSCI 1900

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. CSCI 1900

8. Tautology Example (cont) CSCI 1900

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 CSCI 1900

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 CSCI 1900

11. Show modus ponens = Rule of Inference CSCI 1900

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 CSCI 1900

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 CSCI 1900

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 realIf 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 CSCI 1900

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 happyI am happy I am watching Star Trek • Argument is not valid since its form is: p  qq  p Which is not modus ponens CSCI 1900

16. Invalid Argument (continued) • Truth table shows that this is not a tautology: CSCI 1900

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 CSCI 1900

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 CSCI 1900

19. Another Indirect Method Example • Prove that if the square of an integer is odd, then the integer is odd too. • p: n2 is odd • q: n is odd • ~q  ~p: If n is even, then n2 is even. • If n is even, then there exists an integer m for which n = 2×m. n2 therefore would equal (2×m)2 = 4×m2 which must be even. CSCI 1900

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 CSCI 1900

21. Proof by Contradiction (cont) CSCI 1900

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 CSCI 1900

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 CSCI 1900