discrete structure chapter 2.0 logic 88

8/10/2019 Discrete Structure Chapter 2.0 Logic 88

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Chapter 2: The Fundamentals of Logic

Logic is commonly known as the science ofreasoning.

Logic as a working tool in proving theoremsor solving problems, creativity and insight

are needed, which cannot be taught. Reason to study logic:

Hardware level the design of logic

circuits to implement instruction Software level a knowledge of logic ishelpful in the design of programs.


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Chapter 2:Fundamentals of Logic-Outline

2.1: Logical Form2.2: Truth Tables

2.3: The Law of Logic2.4: Valid and Invalid Arguments2.5: Rule of Inference

2.6: Quantified Statements


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2.1: Logical Form

Basic connectives:Primitive statements(propositions): declarative

sentences that are either True or False ; but not both.

Eg: Susanna wrote Discrete Mathematics book.Eg: 2 + 3 = 5Not statements:

Eg: What a beautiful morning!Eg: Get up and do your exercises.


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2.1: Logical Form

use lowercase letters, such as p, q, r,. torepresent propositions.

Eg: p : It is raining

The truth value of a proposition is true, denoted by

T or 1 whereas the truth value of a proposition isfalse, denoted by F or 0.


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2.1: Logical Form

2 + 3 = 7 X + 1 = 5

3 + 1 Go away! SSK3003 is course code for Discrete Structures

I wear a red shirt 2 + 2 = 4

Proposition with truth value (F)

Not a proposition

Not a proposition

Not a proposition

Proposition with truth value (T)

Proposition with truth value (F)

Proposition with truth value (T)

Exercise:


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2.1: Logical Form

Proposition represented by p, q, r. areconsidered as primitive proposition no way to

break to anything simpler.

Two ways to obtain new proposition:-1.Transform proposition p that is given to p,which denotes its negation and is read NOT

p.

2.Combine two or more propositions intocompound proposition using logicalconnectives.


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2.1: Logical Form

Compound statements: combined primitivestatements by logical connectives or bynegation.

Logical connectives:a) conjunction(AND): p qb) Disjunction(inclusive OR): p q

c) Exclusive OR:d) Implication: p q (if p then q)e) Biconditional: p q (p if only if q , or p iff q)

Logical connectives:


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2.1: Logical Form

Let p and q be propositions. The conjunction of pand q is denoted by p ^ q, which is read p and q

True only both p and q are true and false otherwise.Eg: x : I am a many : I have five children

I am a man and I have five children

conjunction(AND): p q


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2.1: Logical Form

Disjunction(inclusive OR): p q

Disjunction of p and q, is denoted by p v q whichis read p or q.

or is used in inclusive way The proposition is

false only when both p and q are false, otherwise itis true. Sometimes write and/or to point this out. The exclusive or is denoted by p v q.

The compound proposition is true only p or q istrue but not both are true or false.

p : I am a girlq : I am a boy


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2.1: Logical Form

We say p implies q p q Alternatively

If p, then q p is sufficient for q p is a sufficient condition for q

q is necessary for p q is necessary condition for p p only if q

Implication: p q (if p then q)


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2.1: Logical Form

Is denoted by p q or p iff q : which is read if and only if p q (p q) ^ ( q p) Example:

y : I go to school everyday.q : I score A

y q I go to school everyday if and only if I score A

Biconditional: p q (p if only if q , or p iff q)


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2.1: Logical FormEg 1: Negation

p: Combinatorics is a required course for sophomores. p: Combinatorics is not a required course for sophomores.

Eg 2: conjunction(AND)

p: Combinatorics is a required course for sophomores.q: Susanna wrote Discrete Mathematics book.

p q: Combinatorics is a required course for sophomores and Susanna wrote Discrete Mathematics book.


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2.1: Logical FormEg 3: Disjunction(inclusive OR)

p: Combinatorics is a required course for sophomores.

q: Susanna wrote Discrete Mathematics book.

p q: Combinatorics is a required course for sophomores orSusanna wrote Discrete Mathematics book.

Eg 4:Implication(if p then q) p: Combinatorics is a required course for sophomores.q: Susanna wrote Discrete Mathematics book.p q: If Combinatorics is a required course for sophomores

then Susanna wrote Discrete Mathematics book.Note: p is the hypothesis of the implication.Note: q is the conclusion.


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2.1: Logical Form

Eg 3: Biconditional (p if only if q , or p iff q) p:Combinatorics is a required course for sophomores.q: Susanna wrote Discrete Mathematics book.

p q: Combinatorics is a required course forsophomores if and only if Susanna wroteDiscrete Mathematics book.


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2.1: Logical Form

The number x is an integer

Is not a statement because its truth value cannot bedetermined until a numerical value is assignedfor x.


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2.1: Logical Form

Ex 1:

s: David goes out for a walk.t: The moon is out.u: It is snowing.

(t u) s :


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2.1: Logical Form

Ex 2:


(u t) s :


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2.1: Logical Form

Ex 3:


t ( u s) :


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2.1: Logical Form

Ex 4:


(s (u t)) :


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2.1: Logical Form

Translating English Sentences to logicalexpression:

Why?Reasons:a. English (and every other human language) is often

ambiguous. Translating removes the ambiguity. b. Easy to analyze logical expressions to determine their truth

values, easy to manipulate.


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2.1: Logical Form

Ex 5: Translating from English to logicalexpression

Write each of the following sentences symbolically:a. It is not hot but it is sunny.

b. It is neither hot nor sunny.Answer:Let h: It is hot.

s: It is sunny.a. h s

b. h s


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2.1: Logical Form

Ex 6: Searching on the InternetInternet search engines allow you to use some form ofand , or , not to refine the search process.

If you want to find web pages about careers in mathematics orcomputer science but not finance or marketing, how youwant to quote your search?

Ans: Careers AND (mathematics OR computer science) AND NOT (finance OR marketing)


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2.1: Logical Form

Ex 7: And, or and InequalitiesSuppose x is a particular real number. Let p, q and r

symbolize as 0 < x , x < 3 and x = 3respectively.

Write the following inequalities symbolically:

a. x < 3b. 0 < x < 3

c. 0 < x < 3

Ans:

a. q r b. p q c. p (q r)


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2.1: Logical Form

Ex 8: Translate English sentence into a logical

expressionYou can access the Internet from campus only if you are a

computer science major or you are not a freshman. Ans:Let a: You can access the Internet from campus.

c: You are a computer science major.f: You are a freshman

a (c f)


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2.1: Logical Form

Ex 9: Translate English sentence into a logical

expressionYou cannot ride the roller coaster if you are under 4 feet tall

unless you are older than 16 years old. Ans:Let r: You cannot ride the roller coaster.

s: You are under 4 feet tall.q: You are older than 16 years old.

(r s) q


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2.1: Logical Form

Converse, contrapositive and inverse:

p q

The converse of p q is q p

The contrapositive of p q is q p

The inverse of p q is p q


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2.1: Logical Form

Eg: Converse, contrapositive and inverse:

What are the contrapositive, the converse and the inverse ofthe implication

The home team wins whenever it is raining. ? Contrapositive:

If the home team does not win, then it is not raining. Converse: If the home team wins, then it is raining. Inverse: If it is not raining, then the home team does not win.


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2.1: Logical Form

Precedence of Logical operator:

Operator Precedence

1

23

4

5

h bl


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2.2: Truth Tables

p q p q p q p q p q p q

0 0 0 0 0 1 1

0 1 0 1 1 1 0

1 0 0 1 1 0 0

1 1 1 1 0 1 1

A truth table displays the relationship between the

truth values of propositions


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2.2: Truth Tables

Def: A compound statement is called atautology (T 0) if it is true for all truth valueassignments for its component statements.

If a compound statement is false for all such

assignments, then it is called acontradiction(F 0).


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A compound statement is called a tautology if it is true for all truth value assignmentsfor its component statements.

If all false --- contradiction .

p q p v q p (p v q) p p ^ q p ^ ( p ^ q)

0 0 0 1 1 0 0

0 1 1 1 1 1 0

1 0 1 1 0 0 0

1 1 1 1 0 0 0


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2.2: Truth Tables

Logical Equivalence:Def: Two statements forms are called logically equivalent if,

and only if, they have identical truth values for each possible substitution of statements for their statementvariables.

P logically equivalent to Q is denoted by P = Q .


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2.2: Truth Tables

Logical Equivalence:

Eg 1: Show that the propositions p q and p q arelogically equivalent.

p q p p q p q

TTF

F

TFT

F

FFT

T

TFT

T

TFT

T


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2.2: Truth Tables


Eg 2: Show that the propositions p (q r) and (p q) (p r) arelogically equivalent.

p q r q r p (q r) p q p r (p q) (p r)

00

00

01

0 0 0 0 0


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Exercise

1. Verify that [p (q r)] [(p q) (p r)] is atautology.

2. Show that (p ( p q)) and p q are logically

equivalent.

3. Show that (p q) ( p q) is a tautology.


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2.3: Law of Logic

Two propositions p 1 and p 2 are said to be logicallyequivalent and we write p 1 p2 when the

proposition p 1 is true if and only if the p 2 is true.



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2.3: Law of Logic

De Morgan's Laws:

( )

( ) p q p q

p q p q

Note: p and q can be any compound statements.

Augustus De Morgan

1806-1871


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2 3 L f L i


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2.3: Law of Logic

Exercise:

Negate and simplify the compound statement ( ) p q r

[( ) ] [ ( ) ]

[( ) ] ( )

( )

p q r p q r

p q r p q r

p q r


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2.3: Law of Logic

0011

1101

1101

1011

1011

0101

contrapositive of p q

converseinverse

p q p q q p q p p q


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2.3: Law of Logic

Ex: simplification of compound statement

p F pqq p

q pq p

q pq p

q pq p

0

)(

)()(

)()(

)()(

Demorgan's Law

Law of Double Negation

Distributive Law

Inverse Law andIdentity Law


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Ex: Show that (p ( p q)) = ( p q) are logicallyequivalent.

2.3: Law of Logic


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Ex: Show that ( p q) ( p q) is a tautology.

2.3: Law of Logic

2 3 L f L i


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2.3: Law of Logic

Ex: statements : p: Roger studies. q: Roger plays tennis.

r : Roger passes discrete mathematics.premises : p1: If Roger studies, then he will pass discrete math.

p2: If Roger doesn't play tennis, then he'll study. p3: Roger failed discrete mathematics.

Determine whether the argument below is valid

p p r p q p p r

p p p q

p r q p r q

1 2 3

1 2 3

: , : , :

( )

[( ) ( ) ]

which is a tautology,the original argumentis true

( ) p p p q1 2 3

2 3 L f L gi


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2.3: Law of Logic

Def. : If p, q are any arbitrary statements such thatis a tautology, then we say that p logically implies q and we

p q to denote this situation.write

p q means p q is a tautology.

p q means p q is a tautology.


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2.4: Valid & Invalid Arguments

An argument a sequence of statements and are called premises.

Testing an argument for validity:

1. Identify the premises and conclusion of the argument form.2. Construct a truth table showing the truth values of all the premises

and the conclusion.

3. Identify the critical rows:

If all the premises are true and the conclusion is false . Therefore,the argument is invalid .

If all the premises are true and the conclusion is true . Therefore,the argument is valid .


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An Invalid Argument Form:

Eg: Show that the following argument form is invalid.

p q r

q p r

p r

2 4: Valid & Invalid Arguments


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p q r r q r p r p q r

q p r p r

T T T

T T FT F TT F FF T T

F T FF F TF F F

F

TFTF

TFT

T

TFTT

TFT

T

FTFF

FFF

T

TFTT

TTT

T

FTTF

FTT

T

FTFT

TTT

premisesconclusion

From the table, we conclude that this argument form(p q r) ( q p r) ( p r) is invalid .


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Alternatively, from the table below, we conclude that this(p q r) ( q p r) ( p r) is NOT a tautology and

therefore the argument form is invalid

p q r pq r

qp

r

(p q r) ( q

p r)

( p r) (p q r) ( q p r) ( p r)

T T TT T FT F TT F FF T TF T FF F TF F F

TTFTTTTT

TFTTFFTT

TFFTFFTT

TFTFTTTT

TTTFTTTT


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2 5: Rule of Inference


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2.5: Rule of Inference

M odus Ponens :

Syllogism An argument form consisting of two premises and a conclusion.The first and second premises are called the major and minor premises,respectively.

M odus Ponens The most famous form of syllogism in logic.

-(the method of affirming) or the Rule of Detachment

rule of inference - use to validate or invalidate a logical implication withoutresorting to truth table (which will be prohibitively large if the number ofvariables are large).- a form of argument that is valid.- Modus Ponens & modus Tollens are both rule of inference.

[ ( )] p p q q p p q

q



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[ ( )] p p q q p

p q

q

p q p q p q

T TT FF TF F

TFTT

TTFF

TFTF

premisesconclusion

M odus Ponens:



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Eg: M odus Ponens

If the sum of the digits of 371,487 is divisible by 3, then371,487 is divisible by 3.

The sum of the digits of 371,487 is divisible by 3.

371,487 is divisible by 3.

[ ( )] p p q q

p

p q

q



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[( ) ] p q q p

p q

q

p


M odus Tollens - (method of denying) the conclusionis a denial.



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Eg: M odus Tollens

If Zeus is human, the Zeus is mortal.

Zeus is not mortal.

Zeus is not human.

[( ) ] p q q p p q

q

p



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Eg: M odus Ponens

a. Lydia wins a ten million dollar lottery.If Lydia wins a ten million dollar lottery, then Kay willquit her job.

b. If Ali vacations in Paris, then she will have to win a

scholarship.

Ali vacations in Paris.



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Eg: M odus Tollens

a. Lydia wins a ten million dollar lottery.If Lydia wins a ten million dollar lottery, then Kay willquit her job.

b. If Ali vacations in Paris, then he will have to win ascholarship.

Ali vacations in Paris.



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Recognizing M odus Ponens and M odus Tollens:

Use Modus Ponens or modus Tollens to fill in the blanks:-a. If there are more pigeons than there are pigeonholes, then

two pigeons roost in the same hole.

There are more pigeons than there are pigeonholes.

b. If 870,232 is divisible by 6, then it is divisible by 3.870,232 is not divisible by 3.



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Law of the Syllogism:

[( ) ( )] ( ) p q q r p r

p q

q r

p r

Eg:1) If the integer 35244 is divisible by 396, then the integer35244 is divisible by 66.

2) If the integer 35244 is divisible by 66, then the integer35244 is divisible by 3.

3) Therefore, if the integer 35244 is divisible by 396, then theinteger 35244 is divisible by 3.



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p p q

q r

r

2.5: Rule of Inference Law of the Syllogism:

Eg:

1) Rita is baking a cake.

2) If Rita is baking a cake, then she is not practicing herflute.

3) If Rita is not practicing her flute, then her father willnot buy her a car.

4) Therefore Ritas father will not buy her a car.



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p

p q

q r

r

Law of the Syllogism: How to establish the validity of theargument?

Steps: Reasons

1) p q Premise

2) q r Premise

3) p r steps 1 and 2 and the Law of syllogism

4) p Premise

5) r steps 4 and 3 and the Rule of Detachment

Law of the Syllogism: How to establish the validity of the


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p

p q

q r

r

y g yargument?

Steps: Reasons

1) p Premise

2) p q Premise

3) q steps 1 and 2 and the Rule of Detachment

4) q r Premise

5) r steps 3 and 4 and the Rule of Detachment

2 5 R l f I f


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p r r s

t s

t u

u

p

p s

p

Eg: Modus Tollens

s u

p u s t

t u


2 5 R l f I f


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p r r s

t s

t u

u

p

p s

Eg: Modus Tollens(Another reasoning)


t

s p


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Ex : Rule of Conjunction p

q

p q

Ex : Rule of Disjunctive Syllogism

p q

p

q




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Exercise: Establish the validity of the given argument

p q

q (r s)

r ( t u)

p t

u

Steps Reason


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1) p q Premise

2) q (r s) Premise

3) p (r s) Steps 1 and 2 and the Law of Syllogism4) p t Premise

5) p Step 4 and the Rule of Conjunctive Simplification

6) r s Step 5 and 3 and the Rule of Detachment

7) r Step 6 and the Rule of Conjunctive Simplification

8) r ( t u) Premise

9) ( r t) u Step 8, the associative Law of , De Morgans Law

10) t Step 4 and the Rule of Conjunctive Simplification11) r t Step 7 and 10 and the Rule of Conjunction

12) u Steps 9 and 11, the Law of Double Negation,

and the Rule of Disjunctive Syllogism



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Exercise: Establish the validity of the given argument:

If the band could not play rock music or the refreshments werenot delivered on time, then the New Years party would have

been canceled and Alicia would have been angry. If the partywere cancelled, then refunds would have had to be made. Norefunds were made.

Therefore the band could play rock music.

( p q) (r s)

r t

t

u

Steps Reason


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1) r t Premise

2) t Premise

3) r Step 1 and 2 and Modus Tollens4) r s Step 3 and the Rule of Disjunctive

Amplification

5) (r s) Step 4 and De Morgans Law

6) ( p q) (r s) Premise

7) ( p q) Steps 6 and 5 and Modus Tollens

8) p q Step 7, De Morgans Law, the Law ofDouble Negation

9) p Step 8 and the Rule of ConjunctiveSimplification


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The Use of QuantifiersDef. : A declarative sentence is an open statement if(1) it contains one or more variables , and(2) it is not a statement, but(3) it becomes a statement when the variables in it are replaced

by certain allowable choices .

examples: The number x+2 is an even integer. x= y, x> y, x< y, ...

universe



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The Use of Quantifiers

notations: p( x): The number x+2 is an even integer.

q( x, y): The numbers y+2, x- y, and x+2 y are even integers.

p(5): FALSE, p( )7 : TRUE, q(4,2): TRUE

p(6): TRUE, p( )8 : FALSE, q(3,4): FALSE

For some x, p( x) is true.For some x, y, q( x, y) is true.

For some x, is true.For some x, y, is true.

p x( )q x y( , )

Therefore,

2.5: Quantified Statements2.6: Quantified Statements


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existential quantifier : For some x: universal quantifier : For all x:

x

x

x in p( x): free variable x in : bound variable x p x, ( ) x p x, ( ) is either

true or false.



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Ex :

p x x

q x x

r x x x

s x x

( ):

( ):

( ):

( ):

0

0

3 4 0

3 0

2

2

2

x p x r x TRUE

x p x q x TRUE

x p x q x TRUE

x q x s x FALSE

x r x s x FALSE x r x p x FALSE

[ ( ) ( )]:

[ ( ) ( )]:

[ ( ) ( )]:

[ ( ) ( )]:

[ ( ) ( )]:[ ( ) ( )]:

x=4

x=1

x=5,6,... x=-1

universe: real numbers



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Ex : implicit quantification

sin cos2 2 1 x x x x x(sin cos )2 2 1is

"The integer 41 is equal to the sum of two perfect squares."is m n m n[ ]41 2 2



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The Use of QuantifiersDef.: logically equivalent for open statement p( x) and q( x)

x p x q x[ ( ) ( )]

p( x) logically implies q ( x)

x p x q x[ ( ) ( )]

, i.e., p x q x( ) ( ) for any x



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The Use of QuantifiersEx.: Universe: all integers

r x x

s x x

( ):

( ):

2 1 5

92

then x r x s x[ ( ) ( )] is false

but xr x xs x( ) ( ) is true

Therefore, x r x s x[ ( ) ( )] xr x xs x( ) ( )

but x p x q x xp x xq x[ ( ) ( )] [ ( ) ( )]

for any p( x), q( x) and universe



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The Use of QuantifiersFor a prescribed universe and any open statements p( x), q( x):

x p x q x xp x xq x x p x q x xp x xq x

x p x q x xp x xq x

xp x xq x x p x q x

[ ( ) ( )] [ ( ) ( )][ ( ) ( )] [ ( ) ( )]

[ ( ) ( )] [ ( ) ( )]

[ ( ) ( )] [ ( ) ( )] Note this!



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How do we negate quantified statements that involve a singlevariable?

[ ( )] ( )

[ ( )] ( )

[ ( )] ( )

[ ( )] ( )

xp x x p x

xp x x p x

x p x xp x

x p x xp x



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Ex. p( x): x is odd.q( x): x2-1 is even.

Negate x p x q x[ ( ) ( )] (If x is odd, then x2-1 is even.)

[ ( ( ) ( )] [ ( ( ) ( ))]

[ ( ( ) ( ))] [ ( ) ( )]

x p x q x x p x q x

x p x q x x p x q xThere exists an integer x such that x is odd and x2-1 is odd.(a false statement, the original is true)



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multiple variables

x yp x y y xp x y x yp x y y xp x y

( , ) ( , )( , ) ( , )



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BUT

Ex. 2.48 p( x, y): x+ y=17. x yp x y( , ) : For every integer x, there exists an integer y such

that x+ y=17. (TRUE)

y xp x y( , ) : There exists an integer y so that for all integer x, x+ y=17. (FALSE)

),(),( y x xp y y x yp x Therefore,



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Ex

[ [( ( , ) ( , )) ( , )]]

[ [( ( , ) ( , )) ( , )]]

[( ( , ) ( , )) ( , )]

[ [ ( , ) ( , )] ( , )]

[( ( , ) ( , )) ( , )]

x y p x y q x y r x y







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End of Part 1 of Chapter 1

Next Predicate Calculus n Logic Circuits

discrete structure chapter 2.0 logic 88

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