Discrete Mathematics CS 2610

1. Discrete Mathematics CS 2610 August 21, 2008

2. Agenda • Nested quantifiers • Rules of inference • Proofs

3. Review A predicate P, or propositional function, is a function that maps objects in the universe of discourse to propositions • Predicates can be quantified using the universal quantifier (“for all”)  or the existential quantifier (“there exists”)  • Quantified predicates can be negated as follows • x P(x)  x P(x) • x P(x)  x P(x) • Quantified variables are called “bound” • Variables that are not quantified are called “free”

4. Predicate Logic and Propositions • An expression with zero free variables is an actual proposition Ex. Q(x) : x > 0, R(y): y < 10  x Q(x)  y R(y)

5. Nested Quantifiers • When dealing with polyadic predicates, each argument may be quantified with its own quantifier. • Each nested quantifier occurs in the scope of another quantifier. Examples: (L=likes, UoD(x)=kids, UoD(y)=cars) • xy L(x,y) reads x(y L(x,y)) • xy L(x,y) reads x(y L(x,y)) • xy L(x,y) reads x(y L(x,y)) • xy L(x,y) reads x(y L(x,y)) Another example • x (P(x)  y R(x,y))

6. Order matters!!! Examples • If L(x,y) means x likes y, how do you read the following quantified predicates? y L(Alice,y) yx L(x,y) xy L(x,y) x L(x, Prius) Alice likes some car There is a car that is liked by everyone Everyone likes some car Everyone likes the Prius

7. Negation of Nested Quantifiers • To negate a quantifier, move negation to the right, changing quantifiers as you go. Example: xyz P(x,y,z)  x y z P(x,y,z).

8. Proofs and inference Assume that the following statements are true: I have a total score over 96. If I have a total score over 96, then I get an A in the class. What can we claim? I get an A in the class. How do we know the claim is true? Logical Deduction.

9. Proofs • A theorem is a statement that can be proved to be true. • A proof is a sequence of statements that form an argument.

10. Proofs: Inference Rules • An Inference Rule: “” means “therefore” premise 1 premise 2 …  conclusion

11. p p  q  q Proofs: Modus Ponens I have a total score over 96. If I have a total score over 96, then I get an A for the class.  I get an A for this class Tautology: (p  (p  q))  q

12. q p  q  p Proofs: Modus Tollens • If the power supply fails then the lights go out. • The lights are on.  The power supply has not failed. Tautology: (q  (p  q))  p

13. p  p  q Proofs: Addition • I am a student.  I am a student or I am a visitor. Tautology: p  (p  q)

14. p  q p Proofs: Simplification • I am a student and I am a soccer player.  I am a student. Tautology: (p  q) p

15. p q p  q Proofs: Conjunction • I am a student. • I am a soccer player.  I am a student and I am a soccer player. Tautology: ((p)  (q)) p  q

16. p  q q  p Proofs: Disjunctive Syllogism I am a student or I am a soccer player. I am a not soccer player.  I am a student. Tautology: ((p  q)  q)  p

17. p  q q  r  p  r Proofs: Hypothetical Syllogism If I get a total score over 96, I will get an A in the course. If I get an A in the course, I will have a 4.0 semester average.  If I get a total score over 96 then • I will have a 4.0 semester average. Tautology: ((p  q)  (q  r))  (p  r)

18. p  q  p  r  q  r Proofs: Resolution I am taking CS1301 or I am taking CS2610. I am not taking CS1301 or I am taking CS 1302.  I am taking CS2610 or I am taking CS 1302. Tautology: ((p  q )  ( p  r))  (q  r)

19. p  q p  r q  r  r Proofs: Proof by Cases I have taken CS2610 or I have taken CS1301. If I have taken CS2610 then I can register for CS2720 If I have taken CS1301 then I can register for CS2720  I can register for CS2720 Tautology: ((p  q )  (p  r)  (q  r))  r

20. q p  q  p Fallacy of Affirming the Conclusion • If you have the flu then you’ll have a sore throat. You have a sore throat.  You must have the flu. Fallacy: (q  (p  q))  p Abductive, rather than deductive reasoning!

21. p p  q q Fallacy of Denying the Hypothesis • If you have the flu then you’ll have a sore throat. • You do not have the flu.  You do not have a sore throat. Fallacy: (p  (p  q))  q

22. Inference Rules for Quantified Statements Universal Instantiation (for an arbitrary object c from UoD) xP(x)P(c) Universal Generalization (for any arbitrary element c from UoD) P(c)___  xP(x) xP(x)P(c) Existential Instantiation (for some specific object c from UoD, that has not yet occurred!) Existential Generalization (for some specific object c from UoD) P(c)__  xP(x)

23. Vacuous & Trivial Proofs p  q is  vacuously true  if p is false In this case, p  q is a vacuous proof p  q   is trivially true if q is true In this case, we have a trivial proof

24. Proofs • Direct Proof: To prove p  q, we assume p and show/derive q • Indirect Proof by Contraposition: To prove p  q, we prove its contrapositive,  q   p • So assume  q and show/derive  p

25. Proofs Indirect Proof by Contradiction: To prove p, we assume  p and derive a contradiction. Based on the tautology (  p  F )  p “if the negation of p implies a contradiction then p must be true” (aka: Reductio ad Absurdum)

26. More Proofs • Equivalence: To prove p  q, we prove p  q and q  p • Cases: To prove (p1 v p2 v … v pn)  q, we prove (p1  q)  (p2  q)  …  (pn  q)

27. More Proofs Quantifiers: x P(x) : provide a proof, or counterexample. x P(x): Existence Constructive Proof: Find an a in the UoD such that P(a) holds. Existence Non-Constructive Proof: Prove that x P(x) is true without finding an a in the UoD such that P(a) holds