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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:
s: David goes out for a walk.t: The moon is out.u: It is snowing.
(u t) s :
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2.1: Logical Form
Ex 3:
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 4:
s: David goes out for a walk.t: The moon is out.u: It is snowing.
(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
Logical Equivalence:
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.
Logical Equivalence:
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2 3 L f L i
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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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2.4: Valid & Invalid Arguments
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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2.4: Valid & Invalid Arguments
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
2 5: Rule of Inference
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2.5: Rule of Inference
[ ( )] 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:
2 5: Rule of Inference
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2.5: Rule of Inference
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
2 5: Rule of Inference
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[( ) ] p q q p
p q
q
p
2.5: Rule of Inference
M odus Tollens - (method of denying) the conclusionis a denial.
2 5: Rule of Inference
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2.5: Rule of Inference
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
2 5: Rule of Inference
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2.5: Rule of Inference
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.
2.5: Rule of Inference
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2.5: Rule of Inference
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.
2.5: Rule of Inference
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2.5: Rule of Inference
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.
2 5: Rule of Inference
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2.5: Rule of Inference
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.
2.5: Rule of Inference
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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.
2.5: Rule of Inference
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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: Rule of Inference
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)
2.5: Rule of Inference
t
s p
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Ex : Rule of Conjunction p
q
p q
Ex : Rule of Disjunctive Syllogism
p q
p
q
2.5: Rule of Inference
2.5: Rule of Inference
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2.5: Rule of Inference
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
2.5: Rule of Inference
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2.5: Rule of Inference
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
2.6: Quantified Statements
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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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The Use of Quantifiers
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.
2.6: Quantified Statements
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The Use of Quantifiers
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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The Use of Quantifiers
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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The Use of Quantifiers
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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The Use of Quantifiers
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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The Use of Quantifiers
multiple variables
x yp x y y xp x y x yp x y y xp x y
( , ) ( , )( , ) ( , )
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The Use of Quantifiers
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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The Use of Quantifiers
Ex
[ [( ( , ) ( , )) ( , )]]
[ [( ( , ) ( , )) ( , )]]
[( ( , ) ( , )) ( , )]
[ [ ( , ) ( , )] ( , )]
[( ( , ) ( , )) ( , )]
x y p x y q x y r x y
x y p x y q x y r x y
x y p x y q x y r x y
x y p x y q x y r x y
x y p x y q x y r x y
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End of Part 1 of Chapter 1
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