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Discrete Mathematics CS 2610

Discrete Mathematics CS 2610. August 24, 2006. Agenda. Last class Introduction to predicates and quantifiers This class Nested quantifiers Proofs. Overview of last class.

ashely-caldwell
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Discrete Mathematics CS 2610

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  1. Discrete Mathematics CS 2610 August 24, 2006

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

  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. 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 LUV(x, Raymond) Alice likes some car There is a car that is liked by everyone Everyone likes some car Everyone loves Raymond

  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 (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. 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 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) Existential Generalization (for some object c from UoD) P(c)__  xP(x)

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