1. Discrete Mathematics CS 2610 August 24, 2006

2. 2 Agenda Last class Introduction to predicates and quantifiers This class Nested quantifiers Proofs

3. 3 Overview of last class 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. 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. 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. 6 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 LUV(x, Raymond) Alice likes some car There is a car that is liked by everyone Everyone likes some car Everyone loves Raymond Order matters!!!

7. 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. 8 Proofs (or Fun & Games Time) 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? Elementary my dear Watson! Logical Deduction.

9. 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. 10 Proofs: Inference Rules An Inference Rule: “  ” means “therefore” premise 1 premise 2 …  conclusion

11. 11 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 p p  q  q Tautology: (p  (p  q))  q

12. 12 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  q p  q  p p

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

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

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

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

17. 17 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. p  q q  r  p  r Tautology: ((p  q)  (q  r))  (p  r)

18. 18 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. p  q  p  r  q  r Tautology: ((p  q )  (  p  r))  (q  r)

19. 19 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 p  q p  r q  r  r Tautology: ((p  q )  (p  r)  (q  r))  r

20. 20 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 q p  q  p Abductive reasoning

21. 21 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  p p  q  q q

22. 22 Inference Rules for Quantified Statements x P(x)  P(c) x P(x)  P(c) x P(x)  P(c) x P(x)  P(c) P(c)___   x P(x) Universal Instantiation (for an arbitrary object c from UoD) Universal Generalization (for any arbitrary element c from UoD) Existential Instantiation (for some specific object c from UoD) P(c)__   x P(x) Existential Generalization (for some object c from UoD)