CSCI2110 – Discrete Mathematics Tutorial 9 First Order Logic Wong Chung Hoi (Hollis)

Agenda First Order Logic Multiple Quantifiers Proofing Arguments Validity – Proof by truth table – Proof by inference rules

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

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. – –

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

Exercise Express in terms of. – – What is the negation of?

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

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.

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

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.

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.

Agenda First Order Logic Multiple Quantifiers Proofing Arguments Validity – Proof by truth table – Proof by inference rules

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.

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.

Exercise Express in terms of. – – What is the negation of?

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.

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.

Agenda First Order Logic Multiple Quantifiers Proofing Arguments Validity – Proof by truth table – Proof by inference rules

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.

Proof By Truth Table 1 Is this argument valid?

Proof By Truth Table 2 Is this argument valid?

Inference Rules All can be proven by truth table Modus PonensModus Tollens GeneralizationSpecialization TransitivityContradiction Rule

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

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

Inference rule for predicates Universal instantiation Universal Modus Pollens Universal Modus Tollens

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

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

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