1. 1 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Lecture 03: PROOFS Section 1.5 Jarek Rossignac CS1050: Understanding and Constructing Proofs Spring 2006

2. 2 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Questions, concerns, suggestions? Projects –Installing Processing –Creating / posting your PPP –Editing / running your program –Deliverables / due dates Lectures –Posted slides –Reading material –Content of the lectures / Sections Exercises Quizzes Grading

3. 3 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Lecture Objectives Learn how to judge the validity of an argument and how to detect fallacies Learn basic strategies for constructing proofs

4. 4 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Pythagorean theorem? With your team: State the theorem (clarify the assumptions) Develop a proof Write it down on a loose sheet Put names on it –Team name –Member names Pass it forward Let’s hear some solutions… My favorite proof C A B

5. 5 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac A proof of the Pythagorean theorem Assume right triangle with hypotenuse of length C and the other sides of length A and B C A B C AB A B C

6. 6 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Proof details Large square has side length A+B, hence area (A+B) 2. –Prove that the large figure is a square It may be decomposed into 4 triangles (each of area AB/2) and a square (white hole) of area C 2. –Prove that the hole is a square Equate 2 expressions for the area (A+B) 2 = C 2 + 4AB/2 A 2 +2AB+B 2 = C 2 + 2AB A 2 +B 2 = C 2 C A B A B C

7. 7 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac What is a theorem and how is it proven? A theorem is a statement that can be shown to be true. Its proof is an argument comprising a series of statements, derived through methods from previous statements, axioms, or postulates and from previously proven theorems.

8. 8 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac What are lemmas and corollaries? A lemma is a simple theorem used in the proof of another more important theorem. A corollary is a proposition that follows directly from a theorem.

9. 9 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac What is a conjecture? A statement whose truth value is still unknown? –If proven, it will become a theorem.

10. 10 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac What are rules of inference? Rules defining the valid steps that can be used in a proof to justify the next statement. They are the tools of propositional logic. They justify that a conclusion follows logically from a set of hypotheses.

11. 11 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac What is modus ponens? It is a rule of inference based on the tautology (p  (p  q))  q –it is also called the law of detachment  denotes therefore It is snowing If it snows, we go skiing ___________   We go skiing p (Hypothesis 1) p  q (Hypothesis 2) ___________   q (Conclusion)

12. 12 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac What is the list of rules of inference? Addition p  p  q Simplification p  q  p Conjunction (p)  (q)  p  q Modus ponens p  (p  q)  q Modus tollens ¬q  (p  q)  ¬p Hypothetical syllogism (p  q)  (q  r)  (p  r) Disjunctive syllogism (p  q)  ¬p  q Resolution (p  q)  (¬p  r)  (q  r) Resolution, see Exercise 28 in Section 1.2

13. 13 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Valid argument Hypotheses: “If you send me email, I will finish the program. Otherwise I will go to bed early”. “If I go to bed early, I will wake up happy”. Conclusion: “If I do not finish the program, I will wake up happy” p = “you send me email” q = “I finish the program” r = “I go to bed early” s = “I will wake up happy” Develop a valid argument individually. Write it down. Explain it to your team.

14. 14 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Valid argument (solution) Hypotheses: “If you send me email, I will finish the program. Otherwise I will go to bed early”. “If I go to bed early, I will wake up happy”. Conclusion: “If I do not finish the program, I will wake up happy” p = “you send me email”, q = “I finish the program” r = “I go to bed early”, s = “I will wake up happy” 1) p  q Hypothesis (first part) 2) ¬q  ¬p Contrapositive 3) ¬p  r Hypothesis (otherwise of first part) 4) ¬q  r Hypothetical Syllogism from 2) and 3) 5) r  s Hypothesis (second part) 6) ¬q  sHypothetical Syllogism from 4) and 5)

15. 15 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac How is resolution using clauses Hypothesis 1: “Anne is skiing or it is not snowing” Hypothesis 2: “It is snowing or Bart is skating” Conclusion: “Anne is skiing or Bart is skating” Resolution ((p  q)  (¬p  r)  (q  r)) is used in automatic theorem proving. The hypotheses must be clauses: disjunctions of (possibly negated) variables Hence we must convert other statements into clauses, using deMorgan Laws… For example: p  (q  r) replaced by (p  q)  (p  r)

16. 16 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac What is affirming the conclusion? Affirming the conclusion: (p  q)  q  p It is not a tautology! It is a fallacy! H1: “If you do all the exercises (p), then you will get an A (q)” H2: “You got an A (q)” C: “You did all the exercises (p)” Maybe you got an A because you are a genius!

17. 17 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac What is denying the hypothesis? Denying the hypothesis: (p  q)  ¬p  ¬q It is not a tautology! It is a fallacy! H1: “If you do all the exercises (p), then you will get an A (q)” H2: “You did not do all the exercises (¬p)” C : “You did not get an A (¬q)” Maybe you got lucky!

18. 18 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Rules of inference with quantifiers? Universal instantiation: (  x P(x) )  P(c) for any particular c “All women are wise”  “Catherine is wise” Universal generalization: P(c) for an arbitrary c  (  x P(x) ) We did not assume anything specific about c Existential instantiation: (  x P(x) )  there is a c for which P(c) We pick a c and give it a name Existential generalization: P(c) for some c  (  x P(x) )

19. 19 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac What is a direct proof To show that (p  q) prove that p  ¬q is impossible: - Assume p is true - Use rules of inference and theorems to show that - q is true

20. 20 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Example of direct proof “If n is an odd integer, then n 2 is an odd integer” Suppose that n is odd. Then, there is an integer k such that n=2k+1. Hence, n 2 =(2k+1) 2 = 4k 2 +4k+1 = 2(2k 2 +2k) +1 Thus n 2 is odd.

21. 21 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac What is an indirect proof To show that (p  q) prove its contrapositive: ¬q  ¬p - Assume q is false - Use rules of inference and theorems to show that - p is false

22. 22 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Example of indirect proof “If 3n+2 is odd, then n is odd” Suppose that n not odd (i.e., n is even). Then there is an integer k such that n=2k. Hence, 3n+2 = 3(2k)+2 = 2(3k+1) Thus 3n+2 is not odd.

23. 23 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac n integer and n 2 odd  n odd Direct proof does not lead anywhere –  k n 2 =2k+1   m n=2m+1 ? Indirect proof –Assume n is even:  k n=2k –Then n 2 =2(2k 2 ) –Hence n 2 is not odd

24. 24 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac What is a vacuous proof? Prove p  q by showing that p is false (3<0)  (1+1=3) is true since 3<0 is false!!!

25. 25 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac What is a trivial proof? Prove p  q by showing that q is true (1+1=3)  (3>0) is true since (3>0) is true!!!

26. 26 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac What is a proof by contradiction? Prove the p is true by showing that (¬p  false) is true Assume ¬p Prove that leads to something false (contradiction)

27. 27 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Example of proof by contradiction Prove that √2 is irrational Suppose that √2 is rational then  a  Z  b  Z √2 = a/b (Z is the set of integers) assume wlg that a and b have no common factor hence 2 = a 2 / b 2 thus 2 b 2 = a 2 and a 2 is even therefore a is even (contrapositive of n 2 odd  n odd) hence  a  Z a = 2k thus 2 b 2 = a 2 = 4k 2 and b 2 = 2k 2 hence b 2 is even and hence b is even thus a and b are divisible by 2 and have a common factor Contradition! Hence supposition is false and √2 is irrational

28. 28 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac What is a proof by cases? Prove (a  b  c)  q by showing that (a  q)  (b  q)  (c  q) Show that |x||y|=|xy| Case 1 (both positive): xy=xy Case 2 (both negative): (–x)(-–y)=xy Case 3 (opposite sign): (–x)(y)= –xy The other case by symmetry Case 4 (one or both are zero): 0=0

29. 29 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac What is an existance proof? A proof that  x P(x) Often it suffices to find one such an element (constructive proof). We may also use contradiction and show that assuming that no such element exists leads to a contradiction

30. 30 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac What is a uniqueness proof? A proof that  !x P(x) Find such an element and name it x (constructive proof). Then show that x is unique by contradiction: Prove that (y≠x)  P(y) leads to a contradiction

31. 31 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac What is a counterexample? Prove that  x P(x) is false by producing an x for which P(x) is false

32. 32 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Assigned Reading Section 1.5, pp 56-73 (sets)

33. 33 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Practice exercises for quiz 20 p 75 36 p 75 47 p 76 68 p 76 70 p 76

34. 34 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac Check (do not look at your notes) What is a conjecture? Write the modus ponens Write another rule of inference What is denying the hypothesis? What is a direct proof? What is an indirect proof? What is a proof by contradiction? What if you cannot find a proof? statement with unknown truth value (p  (p  q))  q p  q  p (simplification) (p  q)  ¬p  ¬q Assume p, prove q, conclude Assume p, prove q, conclude (p  q) Assume, show, hence Assume ¬q, show ¬p, hence (p  q) Assume Assume ¬p, show contradiction Counterexample?

35. 35 Georgia Tech, IIC, GVU, 2006 MAGIC Lab http://www.gvu.gatech.edu/~jarekJarek Rossignac A miracle? area of the 2 triangles = 10x11 11 10 10 11 area of the 2 triangles = 10x11+1 1/2 1/2 Bring an explanation to the next lecture an a sheet with team name.