20130905170921MTK3013-Chapter1.3 Predicates and Quantifiers

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1.3 PREDICATES AND

QUANTIFIER

MTK3013

DISCRETE STRUCTURES

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Predicates

A predicate is a statement that contains

variables.

Example:

P(x) :x > 3

Q(x,y) :x =y + 3

R(x,y,z) :x +y =z

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Predicates

A predicate becomes a proposition if thevariable(s) contained is(are)

Assigned specific value(s) Quantified

P(x) :x > 3.What are the truth values of

P(4) andP(2)? Q(x,y) :x =y + 3. What are the truth values

ofQ(1,2) and Q(3,0)?

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Quantifiers

Two types of quantifiers

Universal Existential

Universe of discourse - the particular

domain of the variable in a propositional

function

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Universal Quantification

P(x) is true for all values ofx in the universe

of discourse.

x P(x)

for allx,P(x)

for everyx

,P(x)

The variablex is bound by the universal

quantifier, producing a proposition

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Example

U= {all real numbers},P(x):x+1 >x

What is the truth value of x P(x)

U= {all real numbers}, Q(x):x < 2

What is the truth value of x Q(x)

U= {all students in MTK3013}

R(x) :x has an account on Tabung Haji

What does x R(x) mean?

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For universal quantification

P(x) P(x1) P(x2) P(xn)

If the elements in the universe of discourse can belisted, U= {x1, x2, , xn}x P(x)P(x

1

) P(x2

) P(xn

)

Example

U= {positive integers not exceeding 3} andP(x):

x2 < 10 What is the truth value of x P(x)

P(1) ^ P(2) ^ P (3)

T ^ T ^ T

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Existential Quantification

P(x) is truefor somex in the universe ofdiscourse

x P(x) for somex,P(x)

There exists anx such thatP(x)

There is at least onex

such thatP(x)

The variablex is bound by the existentialquantifier, producing a proposition

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Example

U= {all real numbers}, P(x):x> 3 What is the truth value of x P(x)

U= {all real numbers}, Q(x):x=x+ 1 What is the truth value ofx Q(x)

U= {all students in MTK 3013}

R(x) :xhas an account on Tabung Haji What does x R(x) mean?

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For existential quanti f ication

P(x)P(x1) P(x2) P(xn)

If the elements in the universe of discourse

can be listed, U= {x1, x2, , xn}

x P(x) P(x1) P(x2)

P(xn)

Example

U= {positive integers not exceeding 4} andP(x):x2 > 10 What is the truth value ofx P(x)

P(1) v P(2) v P(3) v P(4)

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Binding Variables

Boundvariable: if a variable is quantified

Free variable: Neither bound nor assigned aspecific value

Example: x P(x) x Q(x,y)

Scope of Quantifiers: Part of a logicalexpression to which a quantifier is applied

Example: x (P(x) Q(x)) x R(x)

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Negation of Quantifiers

Distributing a negation operator across a

quantifier changes a universal to an

existential and vice versa.

~x P(x) x ~P(x)

~x P(x) x ~P(x)

Example:

P(x) :x has taken a course in calculus

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Translating from English

Many ways to translate a given sentence

Goal is to produce a logical expression that is

simple and can be easily used in subsequentreasoning

Steps:

Clearly identify the appropriate quantifier(s)

Introduce variable(s) and predicate(s)

Translate using quantifiers, predicates, andlogical operators

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Example

Every student in this class has studied

Discrete Structures

Solution 1

Assume, U = {all students in MTK 3013}

Solution 2

Assume, U = {all people}

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Example

Some student in this class has visited

Mexico

Solution 1

Assume, U = {all students in MTK 3013}

Solution 2

Assume, U = {all people}

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More Example

C(x):x is a COMPUTING student

M(x):x is an MULTIMEDIA student

S(x):x is a smart student

U= {all students in MTK 3013}

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More Example (Cont..)

Everyone is a COMPUTING student.

x C(x)

Nobody is an MULTIMEDIA student.

x ~M(x) or ~x M(x)

All COMPUTING students are smart students.

x [C(x)S(x)]

Some COMPUTING students are smartstudents.

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Use implication or

conjunction? Universal quantifiers usually take implications

All COMPUTING students are smart students.

x[C(x)S(x)] Correct

x[C(x) S(x)] Incorrect

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Use implication or

conjunction? Existential quantifiers usually take

conjunctions

Some COMPUTING students are smart

students.

x [C(x) S(x)] Correctx [C(x) S(x)] Incorrect

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More Example

No COMPUTING student is an MULTIMEDIAstudent.

Ifx is a COMPUTING student, then that student is notan MULTIMEDIA student.

x [C(x) ~M(x)] There does not exist a COMPUTING student who is

also an MULTIMEDIA student.

~x [C(x) M(x)] If any MULTIMEDIA student is a smart student

then he is also a COMPUTING student.x [(M(x) S(x))C(x)]

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(13b) Determine truth value. U={Z}

n (2n = 3n)

(16b) Determine truth value U={R

}n (x2 = -1)

Exercise 17 (Page 47)

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Nested Quantifiers

Quantifiers that occur within the scope of

other quantifiers

Example:

P(x,y): x + y = 0, U={R}

xy P(x,y)

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Quantifications of Two

Variables

For all pair x,y P(x,y).

xy P(x,y) yx P(x,y)

For every x there is a y such that P(x,y).

xy P(x,y)

There is an x such that P(x,y) for all y.

xy P(x,y) There is a pair x,y such that P(x,y).

xy P(x,y) yx P(x,y)

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Translating statements

with nested quantifiers

U= {all real numbers}x y (x + y = y + x)

x y (x + y = 0)x y ( (x > 0) (y < 0) (xy < 0) )

U= {all students in cs2813}

C(x): x has a computerF(x,y): xandyare friends

x ( C(x) y (C(y) F(x,y)) )

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Translating Sentences

U= {all people}

If a person is female and is a parent, then this

person is someones mother.

U= {all integers}

The sum of two positive integers is positive.

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Is the order of quantifiers

important?

If the quantifiers are of the same type, then

order does not matter If the quantifiers are of different types,

then order is important

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Example U={R}

Q(x,y): x+y=0

What are the truth values for

y x Q(x,y) andx y Q(x,y)

y x Q(x,y): There exist at least one y such that forevery real number x, Q(x,y) is true, i.e. x+y=0.

FALSE (not for every, only when y is x).

Butx y Q(x,y): For every real number x, there is a real

number y such that Q(x,y) is true, i.e x+y =0.TRUE (for every x when y is x)

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20130905170921MTK3013-Chapter1.3 Predicates and Quantifiers