PREDICATES AND QUANTIFIERS, Study Guides, Projects, Research of Discrete Mathematics

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Agenda

Propositional Satisfiability Predicates and Quantifiers  Existential Quantifier   Universal Quantifier 

Propositional Satisfiability

A compound proposition is satisfiable if there

is an assignment of truth values to its variables

that makes it true.

When no such assignments exists, that is, when

the compound proposition is false for all

assignments of truth values to its variables, the

compound proposition is unsatisfiable.

Motivating example

Consider the compound proposition: If Zeph is an octopus then Zeph has 8 limbs. Q1: What are the atomic propositions and how do they form this proposition. Q2: Is the proposition true or false? Q3: Why?

Motivating example

A1: Let p = “Zeph is an octopus” and q = “Zeph has 8 limbs”. The compound proposition is represented by p  q. A2: True! A3: Conditional always true when p is false! Q: Why is this not satisfying?

Motivating Example

Recall: A statement is either True or False

Predicates

A Predicate is a sentence depending on

variables which becomes a statement upon

substituting values in the domain.

Ex: P(x): x is a factor of 12 with domain Z+

• Truth Set

{x Є D| P(x)}

Examples

Let Q(x, y) denote the statement “x = y + 3.” What are the truth values of the propositions Q(1, 2) and Q(3, 0)? Let A(c, n) denote the statement “Computer c is connected to network n,” where c is a variable representing a computer and n is a variable representing a network. Suppose that the computer MATH1 is connected to network CAMPUS2, but not to network CAMPUS1. What are the values of A(MATH1, CAMPUS1) and A(MATH1, CAMPUS2)?

PRECONDITIONS AND POSTCONDITIONS The statements that describe valid input are known as preconditions and The conditions that the output should satisfy when the program has run are known as postconditions. temp == x x == y y == temp

The Universal Quantifier

 Conjunction

 x P ( x )  P ( x

1 )  P ( x 2 )  P ( x 3

 What is the truth value of ∀xP (x), where P (x) is the statement “ x 2 < 10 ” and the domain consists of the positive integers not exceeding 4?

Existential Quantifier

 “ x P ( x )” is true when an instance can be found which when plugged in for x makes P ( x ) true  Let P (x) denote the statement “x > 3.” What is the truth value of the quantification ∃xP (x), where the domain consists of all real numbers?  What is the truth value of ∃xP (x), where P (x) is the statement “x 2

10” and the universe of discourse consists of the positive integers not exceeding 4?  {0,1,2,3,4}, P(4)= 16 > 10 T

Example

A: True. For any “exists” we need to find a positive instance. Since x is the first variable in the expression and is “existential”, we need a number that works for all other y, z. Set x = 0 (want to ensure that y - x is not too small). Now for each y we need to find a positive instance z such that y - x ≥ z holds. Plugging in x = 0 we need to satisfy y ≥ z so set z := y. Q: Did we have to set z := y?

Example

A: No. Could also have used the constant z := 0.

Many other valid solutions.

Q: Isn’t it simpler to satisfy 1

 x  y  z ( y - x ≥ z )

by setting x := y and z := 0?

Example

Every real number has an additive inverse

x y (x+y=0)

Example

Let P(x,y) denote “x.y=y.x”, Assume the

domain is the real numbers.

 Is  x  y (x.y=y.x) true?

 Is  y  x ( x.y=y.x ) true?