CSCI2110 – Discrete Mathematics Tutorial 9 First Order Logic

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