1 / 28

CSCI2110 – Discrete Mathematics Tutorial 9 First Order Logic

CSCI2110 – Discrete Mathematics Tutorial 9 First Order Logic. Wong Chung Hoi (Hollis) chwong@cse.cuhk.edu.hk 2-11-2011. Agenda. First Order Logic Multiple Quantifiers Proofing Arguments Validity Proof by truth table Proof by inference rules. First Order Logic.

melita
Télécharger la présentation

CSCI2110 – Discrete Mathematics Tutorial 9 First Order Logic

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. CSCI2110 – Discrete MathematicsTutorial 9First Order Logic Wong Chung Hoi (Hollis) chwong@cse.cuhk.edu.hk 2-11-2011

  2. Agenda • First Order Logic • Multiple Quantifiers • Proofing Arguments Validity • Proof by truth table • Proof by inference rules

  3. First Order Logic • Predicate - Proposition with variables • P(x): x > 0 p: -5 > 0 • H(y): y is smart h: Peter is smart • G(s,t): s is a subset of t g: {1,2} is a subset of Ø • Domain – Set of values that the variables take.

  4. First Order Logic • Predicates takes different truth value on different substituted values. • P(x): x > 0, P(0) = F, P(1) = T • H(y): y is smart, H(“Peter”) = T, H(“John”) = F H(“Paul”) = F, H(“Mary”) = T • Truth set – set of elements that are evaluated True on a predicate.

  5. From Predicates to Propositions • By substitution P(x): x > 0 • p: P(10), p: P(-1) • By quantifiers • For All – • for every, for any, for each, given any, for arbitrary • There Exists – • there is a, we can find a, at least one, for some

  6. Exercise • Express in terms of . • Express in terms of . • What is the negation of ? • What is the negation of ?

  7. P – Set of all people G(x): x grows up All people grow up. Some people never grow up. All people never grow up

  8. S – Set of all things that can be bought E(x): x is expensive these days. Nothing is expensive these days. Something is expensive these days.

  9. S – Set of things to be described. E(x): x can end well. Not everything can end well. Everything can end well.

  10. P – Set of all people R(x): x can read W(x): x can write Some people can’t read and some people can’t write. All people can read or all people can write.

  11. P – Set of all people A – Set of all American C(x): For x, it’s a crutch L(x): For x, it’s a way of life For some people, it’s a crutch and for all American, it’s a way of life. For all people, it’s not a crutch or for some American, it’s not a way of life.

  12. Agenda • First Order Logic • Multiple Quantifiers • Proofing Arguments Validity • Proof by truth table • Proof by inference rules

  13. Multiple Quantifiers • K(x, y): x takes the course y • Domain of x is set of all CSE students (S) • Domain of y is set of all CSE courses (C) • Two quantifiers of the same type can be combined.

  14. Multiple Quantifiers • K(x, y): x takes the course y • Domain of x is set of all CSE students (S) • Domain of y is set of all CSE courses (C) • Two quantifiers of different type cannot be reverse.

  15. Exercise • Express in terms of . • Express in terms of . • What is the negation of ? • What is the negation of ?

  16. S – Set of all posters P – Set of all people M(x, y): x can make y There are some people who can’t make any posters. All people can make some posters.

  17. R – Set of all retards P – Set of all people K(x, y): x know y Everyone knows some retards. There exists someone who don’t know any retards.

  18. Agenda • First Order Logic • Multiple Quantifiers • Proofing Arguments Validity • Proof by truth table • Proof by inference rules

  19. Proofing Arguments Validity • Arguments – hypothesis and conclusion • E.g. • Valid argument: If all hypothesizes are true, then the conclusion is true. • Proof by truth table. • Proof by Inference rules.

  20. Proof By Truth Table 1 • Is this argument valid?

  21. Proof By Truth Table 2 • Is this argument valid?

  22. Inference Rules • All can be proven by truth table • Modus Ponens Modus Tollens • Generalization Specialization • Transitivity Contradiction Rule

  23. Proof By Inference Rules 1 • Show that the argument is valid.

  24. Proof By Inference Rules 2 • Show that the argument is valid.

  25. Inference rule for predicates • Universal instantiation • Universal Modus Pollens • Universal Modus Tollens

  26. Proof By Inference Rules 3 • Show that the argument is valid. Assume the domain of all predicates is a set and .

  27. Proof By Inference Rules 3 • Show that the argument is valid. Assume the domain of all predicates is a set and .

  28. Summary • Difference between predicates and proposition • Quantifiers and negation • Proofing Arguments Validity • Proof by truth table • Proof by inference rules

More Related