predicates & quantifiers

8/3/2019 Predicates & Quantifiers

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Predicate Logic is an extension of Propositional Logic.

It was used to express the meaning

of wide range of statements inmathematics and computer sciencein ways that permit us to reasonand explore relationships betweenobjects

04/28/12 2ACULA, D. (MATH 102C, UST-

ENG'G ICS)



Suppose we have the ff statements:

x+ 7 = 8

y + 3 < 5

Company x was attacked by thehackers.

Note: the above statement can neitherbe true nor false when the values of the variables are not specified.

These statements have TWO PARTS*The variable x, y etc. (subject of thestatement)

*The predicate (refers to a propertythat the sub ect of the statement can04/28/12 3

ACULA, D. (MATH 102C, UST-ENG'G ICS)



x= 8

S: xP: is equal to eight

y + 3 < 5

S: y + 3P: is less than 5

Company x was attacked by thehackers.

S: Company x

P: was attacked by the hackers.

04/28/12 4ACULA, D. (MATH 102C, UST-

ENG'G ICS)



Let P(x) be a statement involving thevariable x and let D be a set. Wecall P a PROPOSITIONAL FUNCTION

(wrt D) if for each x in D, P(x) is aPROPOSITION. We call D theDOMAIN OF DICOURSE.

04/28/12 5ACULA, D. (MATH 102C, UST-

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Let:

P(x) : x + 2x2 is a rational numbers.

D: set of rational numbers

P(x) : The students scored perfect in the test.

D: set of students

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Determine the truth values of the ff.statements

(1)P(x) : 3x + 5 < 3 ; P(2), P(4)

(1)P(x, y) : 2x – 3y = 4 ; P(0, 1) , P(6, 1)

(1)P(x, y, z) : x – y < z ; P(1, 1,1), P(2, -1, 0)

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QUANTIFICATION used to create aproposition from a propositionalfunction. It expresses the extent to

which a predicate is true over arange of elements.

TWO TYPES OF QUANTIFICATION(1)UNIVERSAL QUANTIFICATION

(2)EXISTENTIAL QUANTIFICATION

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Let P(x) be a proposition function with domain of discourse D. The statement for every x, P(x)

is said to be UNIVERSALL QUANTIFIEDSTATEMENT.

The symbol (∀ universal quantifier ) means “forevery or for All”

The statement “for all x, P(x)” can be written asxP(x).∀

The statement for all x, P(x) is TRUE if P(x) is truefor every x in D.

The statement for all x, P(x) is FALSE if P(x) isfalse for at least one x in D.

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(1) Let P(x) “ x + 2 < 2”. What is thetruth value of the quantification

xP(x), where the domain consist∀

of all non-negative integers?

(1) What is the truth value of xP(x),∀

where P(x) is the statement “x2 <9” and the domain consists of thepositive integers not exceeding 3?

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Let P(x) be a proposition function with

domain of discourse D. The statement for some x, P(x) is said to be EXISTENTIALLY QUANTIFIED STATEMENT.

The symbol (∃ existential quantifier ) means“for some or there Exists”.

The statement “for some x, P(x) can bewritten as xP(x).∃

The statement for some x, P(x) is TRUE if P(x) is true for at least one x in D.

The statement for some x, P(x) is FALSE if P(x) is false for every x in D>

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(1) Let P(x) “ x + 2 < 2”. What is thetruth value of the quantification

xP(x), where the domain consist of ∃

all non-negative integers?

(1) What is the truth value of xP(x),∃

where P(x) is the statement “x2 < 9”

and the domain consists of thepositive integers not exceeding 3?

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ENG'G ICS)



UNIQUENESS QUANTIFIER The statement “There exists a unique x

such that P(x) is true” or “there is

exactly one” or “there is one and onlyone” is an example of quantificationusing UNIQUENESS QUANTIFIER.

And this can be written as !xP(x).∃

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The quantifiers and have higher∃ ∀precedence than all logicaloperations.

Example.

(1) The conjunction of xP(x) and∃

Q(x) : ( xP(x)) Q(x) rather than∃ ∧

x(P(x) Q(x)).∃ ∧

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The occurrence of the variable issaid to be BOUND when thequantifier is used on the variable.

The occurrence of the variable thatis not bound by a quantifier is saidto be FREE.

The part of a logical expression towhich a quantifier is applied calledthe SCOPE of quantifier.

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(1) ∀x(4x + 4y <2z)

Bound: x (by universal quantifier)

Free: y & z

(1) ∃x(P(x) Q(x)) xR(x)⇒ ∧∀Bound: all variables

Free: d.n.e.

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Statements involving predicates andquantifiers are LOGICALLY EQUIVALENT if and only if they have

the same truth value no matterwhich predicates are substitutedinto these statements and whichdomain of discourse is used for the

variables in these propositionalfunctions.

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(1) Show that

∀x(P(x) Q(x)) xP(x) Q(x∧ ≡ ∀ ∧

(2)Show that∃x(P(x) Q(x)) xP(x) Q(x)∨ ≡ ∃ ∨

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NEGATION EQUIVALENTSTATEMENT WHEN ISNEGATIONTRUE?

WHEN IS NEGATIONFALSE?

¬ xP(x)∃ ∀x¬P(x) For every x,

P(x) is false

There is an x

for which P(x)is true.

¬ xP(x)∀ ∃x¬P(x) There is anx for whichP(x) isfalse.

P(x) is true forevery x.

Note: The rules for negations for quantifiersare called DE MORGAN’S LAWS FOR

QUANTIFIERS 04/28/12 19ACULA, D. (MATH 102C, UST-

ENG'G ICS)



SHOW THAT¬ x(P(x) Q(x)) x(P(x) ¬Q(x).∀ ⇒ ≡ ∃ ∧

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TWO-PLACE PREDICATES are referred to asrelational predicates, they express a relationbetween two components.

Let P = is easier than

∃xP(x, y): Some x is easier than y.

∀yP(x,y): Every y is easier than x.

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Two quantifiers are NESTED if one quantifier iswithin the scope of the other quantifier.

EXAMPLE

∀x y((x<0) (y<0)) (xy < 0)

∀ ∨ ⇒

Consider that the domain of discourse for bothvariables are real numbers.

“For all real number x and for all real number

y, if x is less than 0 or y less than 0, then xyis less than 0.

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It is important to note that the order of the

quantifier important, unless all thequantifiers are universal(existential)quantifiers.

Let P(x, y): 2xy = 3x + y

What are the truth values of

∀x yP(x, y)∀ y xP(x, y)∀ ∀

∀x yP(x, y)∃ x yP(x, y)∃ ∀

∃x yP(x, y)∃ y xP(x, y)∃ ∃Where the domain for all variables consists of

all real numbers

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STATEMENT WHEN TRUE? WHEN FALSE?

∀x yP(x,∀

y)∀y xP(x,∀

y)

P(x, y) is true forevery pair x, y.

There is a pair x, y forwhich P(x, y) is false.

∀x yP(x,∃

y)For every x, thereis a y for which P(x,

y) is true.

There is an x such thatP(x, y) is false for

every y.∃x yP(x,∀

y)There is an x forwhich P(x, y) istrue for every y.

For every x there is a yfor which P(x, y) isfalse.

∃x yP(x,∃

y)There is pair x, yfor which P(x, y) is

P(x, y) is false forevery x, y.04/28/12 24




Discrete Math Book

04/28/12 25ACULA, D. (MATH 102C, UST-

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