SlideShare ist ein Scribd-Unternehmen logo
1 von 7
Downloaden Sie, um offline zu lesen
SUBTOPIC 3             :        QUANTIFIERS



       The statement

                                     P:         n is odd integer.

       A proposition is a statement that is either true or false. The statement p is not proposition
because whether p is true or false depends on the value of n. For example, p is true if n = 104 and
false if n = 8. Since, most of the statements in mathematics and computer a science use variable,
we must extend the system of logic to include such statements.

     1.Quantifiers

     Definition:

     Let P (x) be a statement involving the variable x and let D be a set. We call P a proportional
     function or predicate (with respect to D ) , if for each x ∈ D , P (x) is a proposition. We call
     D the domain of discourse of P.



Example 1:     Let P(n) be the statement

                                           n is an odd integer

       Then P is a propositional function with the domain of discourse             since for each n ∈

   , P(n) is a proposition [for each n ∈      , P(n) is true or false but not both]. For example, if n =
1, we obtain the proposition.

                                    P (1): 1 is an odd integer

(Which is true) If n = 2, we obtain the proposition/

                                    P (2): 2 is an odd integer

(Which is false)


                                                                                                     14
A propositional function P, by itself, is neither true nor false. However, for each x is
domain of discourse, P (x) is a proposition and is, therefore, either true or false. We can think of
propositional function as defining a class of propositions, one for each element in the domain of
discourse. For example, if P is a propositional function with domain of discourse         , we obtain
the class of propositions.

                                           P (1), P (2), …..

Each of P (1), P (2), …. Is either true or false.




        2. Universal Quantification

        Definition:

        Let P be a propositional function with the domain of discourse D. The universal
        quantification of P (x) is the statement. “For all values of x, P is true.”




                                               ∀x, P (x)

Similar expressions:

    -     For each…
    -     For every…
    -     For any…


   3. Counterexample

   A counterexample is an example chosen to show that a universal statement is FALSE.

   To verify:

         -   ∀x, P (x) is true
         -   ∀x, P (x) is false




                                                                                                  15
Example 1:

Consider the universally quantified statement.

                                            ∀x (      ≥ 0)

The domain of discourse is R. The statement is true because for every real number x, it is true
that the square of x is positive or zero.

        According the definition, the universally quantified statement.

                                             ∀x, P (x)

Is false for at least one x in the domain of discourse that makes P (x) is false. A value x in the
domain of discourse that makes P (x) false is called a counterexample to the statement.

Example 2:

Consider the universally quantified statement.

                                            ∀x (     -1 ≥ 0)

The domain of discourse is R. The statement is false since, if x = 1, the proposition

                                                   -1 > 0

Is false. The value1 is counterexample of the statement.

                                            ∀x (     -1 ≥ 0)

Although there are values of x that make the propositional function true, the counterexample
provide show that the universally quantified statement is false.




                                                                                               16
4. Existential Quantification

       Let P be a proportional function with the domain of discourse D. The existential
       quantification of P (x) is the statement. “There exists a value of x for which P (x) is true.

                                                   ∃x, P(x)

       Similar expressions:

           -   There is some…
           -   There exist…

       There is at least…




Example 1:

Consider the existentially quantified statement.


∃x (


The domain of discourse is R. the statement is true because it is possible to find at least one real
number x for which the proposition




Is true. For example, if x = 2, we obtain the true proposition.




                                                                                                       17
Is not the case that every value of x results in a true proposition. For example, if x = 1, the
proposition




Is false.

        According to definition, the existentially quantified statement

                                              ∃x, P(x)

Is false for every x in the domain of discourse, the proposition P (x) is false.




                                                                                            18
5. De Morgan’s Law for Logic

      Theorem:

      (∀x, P (x)) ≡ (∃x, (P(x))

      (∃x, (P(x)) ≡ (∀x, P (x))

      The statement

      “The sum of any two positive real numbers is positive”.

      ∀x > 0∀y > 0,


Example 1:        Let P(x) be the statement




We show that

                                               ∃x, P(x)

Is false by verifying that

                                              ∀x, ⌐ P (x)

Is true.

           The technique can be justified by appealing to theorem. After we prove that proposition
is true, we may negate and conclude that is false. By theorem,

                                              ∃x, ⌐⌐P(x)

Or equivalently

                                               ∃x, P(x)

Is also false.


                                                                                               19
1. EXERCISE.


     a) let P (x) be the propositional function “x ≥   .” Tell whether each proposition is
        true or false. The domain of discourse is R
           i.   P (1)

          ii.   ⌐∃x P(x)




     b) Suppose that the domain of discourse of the propositional function P is {1, 2, 3,
        4}. Rewrite each propositional function using only negation, disjunction and
        conjunction.

           i.   ∀x P (x)




     c) Determine the truth value, the domain discourse R x R, justify the answer.

           i.   ∀x ∀y (    < y + 1)




     d) Assume that ∀x ∃y P(x, y) is false and that the domain of discourse is nonempty.
        It must also be false? Prove the answer.

           i.   ∀x ∀y P (x,y)




                                                                                       20

Weitere ähnliche Inhalte

Was ist angesagt?

A Theory of the Learnable; PAC Learning
A Theory of the Learnable; PAC LearningA Theory of the Learnable; PAC Learning
A Theory of the Learnable; PAC Learningdhruvgairola
 
Boolean Programs and Quantified Propositional Proof System -
Boolean Programs and Quantified Propositional Proof System - Boolean Programs and Quantified Propositional Proof System -
Boolean Programs and Quantified Propositional Proof System - Michael Soltys
 
Metalogic: The non-algorithmic side of the mind
Metalogic: The non-algorithmic side of the mindMetalogic: The non-algorithmic side of the mind
Metalogic: The non-algorithmic side of the mindHaskell Lambda
 
P, NP and NP-Complete, Theory of NP-Completeness V2
P, NP and NP-Complete, Theory of NP-Completeness V2P, NP and NP-Complete, Theory of NP-Completeness V2
P, NP and NP-Complete, Theory of NP-Completeness V2S.Shayan Daneshvar
 
Mth3101 Advanced Calculus Chapter 1
Mth3101 Advanced Calculus Chapter 1Mth3101 Advanced Calculus Chapter 1
Mth3101 Advanced Calculus Chapter 1saya efan
 
5.6.08 Fundamental Theorem Of Algebra1
5.6.08   Fundamental Theorem Of Algebra15.6.08   Fundamental Theorem Of Algebra1
5.6.08 Fundamental Theorem Of Algebra1chrismac47
 
Lecture 2 predicates quantifiers and rules of inference
Lecture 2 predicates quantifiers and rules of inferenceLecture 2 predicates quantifiers and rules of inference
Lecture 2 predicates quantifiers and rules of inferenceasimnawaz54
 
A Machine-Assisted Proof of Gödel's Incompleteness Theorems
A Machine-Assisted Proof of Gödel's Incompleteness TheoremsA Machine-Assisted Proof of Gödel's Incompleteness Theorems
A Machine-Assisted Proof of Gödel's Incompleteness TheoremsLawrence Paulson
 

Was ist angesagt? (19)

Per3 logika&amp;pembuktian
Per3 logika&amp;pembuktianPer3 logika&amp;pembuktian
Per3 logika&amp;pembuktian
 
A Theory of the Learnable; PAC Learning
A Theory of the Learnable; PAC LearningA Theory of the Learnable; PAC Learning
A Theory of the Learnable; PAC Learning
 
Lesson 5: Continuity
Lesson 5: ContinuityLesson 5: Continuity
Lesson 5: Continuity
 
Discrete Math Lecture 02: First Order Logic
Discrete Math Lecture 02: First Order LogicDiscrete Math Lecture 02: First Order Logic
Discrete Math Lecture 02: First Order Logic
 
3 fol examples v2
3 fol examples v23 fol examples v2
3 fol examples v2
 
Boolean Programs and Quantified Propositional Proof System -
Boolean Programs and Quantified Propositional Proof System - Boolean Programs and Quantified Propositional Proof System -
Boolean Programs and Quantified Propositional Proof System -
 
Lec17
Lec17Lec17
Lec17
 
Metalogic: The non-algorithmic side of the mind
Metalogic: The non-algorithmic side of the mindMetalogic: The non-algorithmic side of the mind
Metalogic: The non-algorithmic side of the mind
 
Review
ReviewReview
Review
 
P, NP and NP-Complete, Theory of NP-Completeness V2
P, NP and NP-Complete, Theory of NP-Completeness V2P, NP and NP-Complete, Theory of NP-Completeness V2
P, NP and NP-Complete, Theory of NP-Completeness V2
 
Mth3101 Advanced Calculus Chapter 1
Mth3101 Advanced Calculus Chapter 1Mth3101 Advanced Calculus Chapter 1
Mth3101 Advanced Calculus Chapter 1
 
Analysis of algorithms
Analysis of algorithmsAnalysis of algorithms
Analysis of algorithms
 
Math
MathMath
Math
 
Chapter 4
Chapter 4Chapter 4
Chapter 4
 
5.6.08 Fundamental Theorem Of Algebra1
5.6.08   Fundamental Theorem Of Algebra15.6.08   Fundamental Theorem Of Algebra1
5.6.08 Fundamental Theorem Of Algebra1
 
Limits BY ATC
Limits BY ATCLimits BY ATC
Limits BY ATC
 
C2.0 propositional logic
C2.0 propositional logicC2.0 propositional logic
C2.0 propositional logic
 
Lecture 2 predicates quantifiers and rules of inference
Lecture 2 predicates quantifiers and rules of inferenceLecture 2 predicates quantifiers and rules of inference
Lecture 2 predicates quantifiers and rules of inference
 
A Machine-Assisted Proof of Gödel's Incompleteness Theorems
A Machine-Assisted Proof of Gödel's Incompleteness TheoremsA Machine-Assisted Proof of Gödel's Incompleteness Theorems
A Machine-Assisted Proof of Gödel's Incompleteness Theorems
 

Andere mochten auch

Ethnomathematics in School Curriculum
Ethnomathematics in School CurriculumEthnomathematics in School Curriculum
Ethnomathematics in School CurriculumSantosh Bhandari
 
The Concept of Beauty among Makonde sculptors: an ethnomathematical research
The Concept of Beauty among Makonde sculptors: an ethnomathematical research The Concept of Beauty among Makonde sculptors: an ethnomathematical research
The Concept of Beauty among Makonde sculptors: an ethnomathematical research ICEM-4
 
Ethnomodeling
EthnomodelingEthnomodeling
EthnomodelingICEM-4
 
An ethnomathematics study at the workplace: masons’ professional practices
An ethnomathematics study at the workplace: masons’ professional practicesAn ethnomathematics study at the workplace: masons’ professional practices
An ethnomathematics study at the workplace: masons’ professional practicesICEM-4
 
Mya 2010 lunar calendar web version final
Mya 2010 lunar calendar web version finalMya 2010 lunar calendar web version final
Mya 2010 lunar calendar web version finalICEM-4
 
Urban Ethnomathematics and Ethnogenesis: Community Projects in Caparica
Urban Ethnomathematics and Ethnogenesis: Community Projects in CaparicaUrban Ethnomathematics and Ethnogenesis: Community Projects in Caparica
Urban Ethnomathematics and Ethnogenesis: Community Projects in CaparicaICEM-4
 
From Ethnomathematics to Ethnocomputing
From Ethnomathematics to EthnocomputingFrom Ethnomathematics to Ethnocomputing
From Ethnomathematics to EthnocomputingICEM-4
 
ETHNOMATHEMATICS AND ADULT STUDENTS: CHALLENGES TO TEACHERS´CONTINUING EDUCATION
ETHNOMATHEMATICS AND ADULT STUDENTS: CHALLENGES TO TEACHERS´CONTINUING EDUCATIONETHNOMATHEMATICS AND ADULT STUDENTS: CHALLENGES TO TEACHERS´CONTINUING EDUCATION
ETHNOMATHEMATICS AND ADULT STUDENTS: CHALLENGES TO TEACHERS´CONTINUING EDUCATIONICEM-4
 
On the ethnomathematics � epistemology nexus
On the ethnomathematics � epistemology nexusOn the ethnomathematics � epistemology nexus
On the ethnomathematics � epistemology nexusICEM-4
 
An account of the construction of mathematical knowledge in uril, a Capeverde...
An account of the construction of mathematical knowledge in uril, a Capeverde...An account of the construction of mathematical knowledge in uril, a Capeverde...
An account of the construction of mathematical knowledge in uril, a Capeverde...ICEM-4
 

Andere mochten auch (12)

Ethnomathematics in School Curriculum
Ethnomathematics in School CurriculumEthnomathematics in School Curriculum
Ethnomathematics in School Curriculum
 
The Concept of Beauty among Makonde sculptors: an ethnomathematical research
The Concept of Beauty among Makonde sculptors: an ethnomathematical research The Concept of Beauty among Makonde sculptors: an ethnomathematical research
The Concept of Beauty among Makonde sculptors: an ethnomathematical research
 
Mathematics as an issue
Mathematics as an issueMathematics as an issue
Mathematics as an issue
 
Ethnomodeling
EthnomodelingEthnomodeling
Ethnomodeling
 
An ethnomathematics study at the workplace: masons’ professional practices
An ethnomathematics study at the workplace: masons’ professional practicesAn ethnomathematics study at the workplace: masons’ professional practices
An ethnomathematics study at the workplace: masons’ professional practices
 
Mya 2010 lunar calendar web version final
Mya 2010 lunar calendar web version finalMya 2010 lunar calendar web version final
Mya 2010 lunar calendar web version final
 
Urban Ethnomathematics and Ethnogenesis: Community Projects in Caparica
Urban Ethnomathematics and Ethnogenesis: Community Projects in CaparicaUrban Ethnomathematics and Ethnogenesis: Community Projects in Caparica
Urban Ethnomathematics and Ethnogenesis: Community Projects in Caparica
 
From Ethnomathematics to Ethnocomputing
From Ethnomathematics to EthnocomputingFrom Ethnomathematics to Ethnocomputing
From Ethnomathematics to Ethnocomputing
 
ETHNOMATHEMATICS AND ADULT STUDENTS: CHALLENGES TO TEACHERS´CONTINUING EDUCATION
ETHNOMATHEMATICS AND ADULT STUDENTS: CHALLENGES TO TEACHERS´CONTINUING EDUCATIONETHNOMATHEMATICS AND ADULT STUDENTS: CHALLENGES TO TEACHERS´CONTINUING EDUCATION
ETHNOMATHEMATICS AND ADULT STUDENTS: CHALLENGES TO TEACHERS´CONTINUING EDUCATION
 
On the ethnomathematics � epistemology nexus
On the ethnomathematics � epistemology nexusOn the ethnomathematics � epistemology nexus
On the ethnomathematics � epistemology nexus
 
An account of the construction of mathematical knowledge in uril, a Capeverde...
An account of the construction of mathematical knowledge in uril, a Capeverde...An account of the construction of mathematical knowledge in uril, a Capeverde...
An account of the construction of mathematical knowledge in uril, a Capeverde...
 
Slide subtopic 2
Slide subtopic 2Slide subtopic 2
Slide subtopic 2
 

Ähnlich wie Chapter 3

CSci102_Module 4.ppt
CSci102_Module 4.pptCSci102_Module 4.ppt
CSci102_Module 4.pptHarleyGotardo
 
CMSC 56 | Lecture 3: Predicates & Quantifiers
CMSC 56 | Lecture 3: Predicates & QuantifiersCMSC 56 | Lecture 3: Predicates & Quantifiers
CMSC 56 | Lecture 3: Predicates & Quantifiersallyn joy calcaben
 
Chapter 01 - p2.pdf
Chapter 01 - p2.pdfChapter 01 - p2.pdf
Chapter 01 - p2.pdfsmarwaneid
 
1606751772-ds-lecture-6.ppt
1606751772-ds-lecture-6.ppt1606751772-ds-lecture-6.ppt
1606751772-ds-lecture-6.pptTejasAditya2
 
Discreate structure presentation introduction
Discreate structure presentation introductionDiscreate structure presentation introduction
Discreate structure presentation introductionyashirraza123
 
Discrete Structure Lecture #5 & 6.pdf
Discrete Structure Lecture #5 & 6.pdfDiscrete Structure Lecture #5 & 6.pdf
Discrete Structure Lecture #5 & 6.pdfMuhammadUmerIhtisham
 
Quantifiers and its Types
Quantifiers and its TypesQuantifiers and its Types
Quantifiers and its TypesHumayunNaseer4
 
Nested Quantifiers.pptx
Nested Quantifiers.pptxNested Quantifiers.pptx
Nested Quantifiers.pptxJeevan225779
 
Fuzzy logic and application in AI
Fuzzy logic and application in AIFuzzy logic and application in AI
Fuzzy logic and application in AIIldar Nurgaliev
 
Unit 1 quantifiers
Unit 1  quantifiersUnit 1  quantifiers
Unit 1 quantifiersraksharao
 
logicproof-141212042039-conversion-gate01.pdf
logicproof-141212042039-conversion-gate01.pdflogicproof-141212042039-conversion-gate01.pdf
logicproof-141212042039-conversion-gate01.pdfPradeeshSAI
 
20220818151924_PPT01 - The Logic of Compound and Quantitative Statement.pptx
20220818151924_PPT01 - The Logic of Compound and Quantitative Statement.pptx20220818151924_PPT01 - The Logic of Compound and Quantitative Statement.pptx
20220818151924_PPT01 - The Logic of Compound and Quantitative Statement.pptxssuser92109d
 
Welcome to International Journal of Engineering Research and Development (IJERD)
Welcome to International Journal of Engineering Research and Development (IJERD)Welcome to International Journal of Engineering Research and Development (IJERD)
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
 

Ähnlich wie Chapter 3 (20)

Slide subtopic 3
Slide subtopic 3Slide subtopic 3
Slide subtopic 3
 
CSci102_Module 4.ppt
CSci102_Module 4.pptCSci102_Module 4.ppt
CSci102_Module 4.ppt
 
CMSC 56 | Lecture 3: Predicates & Quantifiers
CMSC 56 | Lecture 3: Predicates & QuantifiersCMSC 56 | Lecture 3: Predicates & Quantifiers
CMSC 56 | Lecture 3: Predicates & Quantifiers
 
Per3 logika
Per3 logikaPer3 logika
Per3 logika
 
Formal Logic - Lesson 8 - Predicates and Quantifiers
Formal Logic - Lesson 8 - Predicates and QuantifiersFormal Logic - Lesson 8 - Predicates and Quantifiers
Formal Logic - Lesson 8 - Predicates and Quantifiers
 
Chapter 01 - p2.pdf
Chapter 01 - p2.pdfChapter 01 - p2.pdf
Chapter 01 - p2.pdf
 
1606751772-ds-lecture-6.ppt
1606751772-ds-lecture-6.ppt1606751772-ds-lecture-6.ppt
1606751772-ds-lecture-6.ppt
 
Discreate structure presentation introduction
Discreate structure presentation introductionDiscreate structure presentation introduction
Discreate structure presentation introduction
 
Discrete Structure Lecture #5 & 6.pdf
Discrete Structure Lecture #5 & 6.pdfDiscrete Structure Lecture #5 & 6.pdf
Discrete Structure Lecture #5 & 6.pdf
 
PredicateLogic (1).ppt
PredicateLogic (1).pptPredicateLogic (1).ppt
PredicateLogic (1).ppt
 
PredicateLogic.pptx
PredicateLogic.pptxPredicateLogic.pptx
PredicateLogic.pptx
 
Quantifier
QuantifierQuantifier
Quantifier
 
Quantifiers and its Types
Quantifiers and its TypesQuantifiers and its Types
Quantifiers and its Types
 
Nested Quantifiers.pptx
Nested Quantifiers.pptxNested Quantifiers.pptx
Nested Quantifiers.pptx
 
Predicate &amp; quantifier
Predicate &amp; quantifierPredicate &amp; quantifier
Predicate &amp; quantifier
 
Fuzzy logic and application in AI
Fuzzy logic and application in AIFuzzy logic and application in AI
Fuzzy logic and application in AI
 
Unit 1 quantifiers
Unit 1  quantifiersUnit 1  quantifiers
Unit 1 quantifiers
 
logicproof-141212042039-conversion-gate01.pdf
logicproof-141212042039-conversion-gate01.pdflogicproof-141212042039-conversion-gate01.pdf
logicproof-141212042039-conversion-gate01.pdf
 
20220818151924_PPT01 - The Logic of Compound and Quantitative Statement.pptx
20220818151924_PPT01 - The Logic of Compound and Quantitative Statement.pptx20220818151924_PPT01 - The Logic of Compound and Quantitative Statement.pptx
20220818151924_PPT01 - The Logic of Compound and Quantitative Statement.pptx
 
Welcome to International Journal of Engineering Research and Development (IJERD)
Welcome to International Journal of Engineering Research and Development (IJERD)Welcome to International Journal of Engineering Research and Development (IJERD)
Welcome to International Journal of Engineering Research and Development (IJERD)
 

Mehr von Eli Lilly and Company (17)

Slide subtopic 2
Slide subtopic 2Slide subtopic 2
Slide subtopic 2
 
Slide subtopic 1
Slide subtopic 1Slide subtopic 1
Slide subtopic 1
 
Slide subtopic 4
Slide subtopic 4Slide subtopic 4
Slide subtopic 4
 
Slide subtopic 5
Slide subtopic 5Slide subtopic 5
Slide subtopic 5
 
Chapter 1
Chapter 1Chapter 1
Chapter 1
 
Chapter 1
Chapter 1Chapter 1
Chapter 1
 
Chapter 5
Chapter 5Chapter 5
Chapter 5
 
Akses dan ekuiti
Akses dan ekuitiAkses dan ekuiti
Akses dan ekuiti
 
Isu profesional guru dalam pengajaran matematik
Isu profesional guru dalam pengajaran matematikIsu profesional guru dalam pengajaran matematik
Isu profesional guru dalam pengajaran matematik
 
kurikulum dan kelainan upaya
kurikulum dan kelainan upayakurikulum dan kelainan upaya
kurikulum dan kelainan upaya
 
jantina dan bangsa
jantina dan bangsajantina dan bangsa
jantina dan bangsa
 
jantina dan bangsa
jantina dan bangsajantina dan bangsa
jantina dan bangsa
 
Jantina dan bangsa
Jantina dan bangsaJantina dan bangsa
Jantina dan bangsa
 
ISU JANTINA
ISU JANTINAISU JANTINA
ISU JANTINA
 
Qayyum
QayyumQayyum
Qayyum
 
Trend kurikulum
Trend kurikulumTrend kurikulum
Trend kurikulum
 
Isu profesional guru dalam pengajaran matematik
Isu profesional guru dalam pengajaran matematikIsu profesional guru dalam pengajaran matematik
Isu profesional guru dalam pengajaran matematik
 

Kürzlich hochgeladen

What is the Future of QuickBooks DeskTop?
What is the Future of QuickBooks DeskTop?What is the Future of QuickBooks DeskTop?
What is the Future of QuickBooks DeskTop?TechSoup
 
Patterns of Written Texts Across Disciplines.pptx
Patterns of Written Texts Across Disciplines.pptxPatterns of Written Texts Across Disciplines.pptx
Patterns of Written Texts Across Disciplines.pptxMYDA ANGELICA SUAN
 
The Singapore Teaching Practice document
The Singapore Teaching Practice documentThe Singapore Teaching Practice document
The Singapore Teaching Practice documentXsasf Sfdfasd
 
In - Vivo and In - Vitro Correlation.pptx
In - Vivo and In - Vitro Correlation.pptxIn - Vivo and In - Vitro Correlation.pptx
In - Vivo and In - Vitro Correlation.pptxAditiChauhan701637
 
Philosophy of Education and Educational Philosophy
Philosophy of Education  and Educational PhilosophyPhilosophy of Education  and Educational Philosophy
Philosophy of Education and Educational PhilosophyShuvankar Madhu
 
Education and training program in the hospital APR.pptx
Education and training program in the hospital APR.pptxEducation and training program in the hospital APR.pptx
Education and training program in the hospital APR.pptxraviapr7
 
How to Manage Cross-Selling in Odoo 17 Sales
How to Manage Cross-Selling in Odoo 17 SalesHow to Manage Cross-Selling in Odoo 17 Sales
How to Manage Cross-Selling in Odoo 17 SalesCeline George
 
Prescribed medication order and communication skills.pptx
Prescribed medication order and communication skills.pptxPrescribed medication order and communication skills.pptx
Prescribed medication order and communication skills.pptxraviapr7
 
Benefits & Challenges of Inclusive Education
Benefits & Challenges of Inclusive EducationBenefits & Challenges of Inclusive Education
Benefits & Challenges of Inclusive EducationMJDuyan
 
Practical Research 1: Lesson 8 Writing the Thesis Statement.pptx
Practical Research 1: Lesson 8 Writing the Thesis Statement.pptxPractical Research 1: Lesson 8 Writing the Thesis Statement.pptx
Practical Research 1: Lesson 8 Writing the Thesis Statement.pptxKatherine Villaluna
 
How to Add Existing Field in One2Many Tree View in Odoo 17
How to Add Existing Field in One2Many Tree View in Odoo 17How to Add Existing Field in One2Many Tree View in Odoo 17
How to Add Existing Field in One2Many Tree View in Odoo 17Celine George
 
How to Solve Singleton Error in the Odoo 17
How to Solve Singleton Error in the  Odoo 17How to Solve Singleton Error in the  Odoo 17
How to Solve Singleton Error in the Odoo 17Celine George
 
Easter in the USA presentation by Chloe.
Easter in the USA presentation by Chloe.Easter in the USA presentation by Chloe.
Easter in the USA presentation by Chloe.EnglishCEIPdeSigeiro
 
Patient Counselling. Definition of patient counseling; steps involved in pati...
Patient Counselling. Definition of patient counseling; steps involved in pati...Patient Counselling. Definition of patient counseling; steps involved in pati...
Patient Counselling. Definition of patient counseling; steps involved in pati...raviapr7
 
UKCGE Parental Leave Discussion March 2024
UKCGE Parental Leave Discussion March 2024UKCGE Parental Leave Discussion March 2024
UKCGE Parental Leave Discussion March 2024UKCGE
 
Diploma in Nursing Admission Test Question Solution 2023.pdf
Diploma in Nursing Admission Test Question Solution 2023.pdfDiploma in Nursing Admission Test Question Solution 2023.pdf
Diploma in Nursing Admission Test Question Solution 2023.pdfMohonDas
 
2024.03.23 What do successful readers do - Sandy Millin for PARK.pptx
2024.03.23 What do successful readers do - Sandy Millin for PARK.pptx2024.03.23 What do successful readers do - Sandy Millin for PARK.pptx
2024.03.23 What do successful readers do - Sandy Millin for PARK.pptxSandy Millin
 

Kürzlich hochgeladen (20)

What is the Future of QuickBooks DeskTop?
What is the Future of QuickBooks DeskTop?What is the Future of QuickBooks DeskTop?
What is the Future of QuickBooks DeskTop?
 
Personal Resilience in Project Management 2 - TV Edit 1a.pdf
Personal Resilience in Project Management 2 - TV Edit 1a.pdfPersonal Resilience in Project Management 2 - TV Edit 1a.pdf
Personal Resilience in Project Management 2 - TV Edit 1a.pdf
 
Patterns of Written Texts Across Disciplines.pptx
Patterns of Written Texts Across Disciplines.pptxPatterns of Written Texts Across Disciplines.pptx
Patterns of Written Texts Across Disciplines.pptx
 
The Singapore Teaching Practice document
The Singapore Teaching Practice documentThe Singapore Teaching Practice document
The Singapore Teaching Practice document
 
In - Vivo and In - Vitro Correlation.pptx
In - Vivo and In - Vitro Correlation.pptxIn - Vivo and In - Vitro Correlation.pptx
In - Vivo and In - Vitro Correlation.pptx
 
Philosophy of Education and Educational Philosophy
Philosophy of Education  and Educational PhilosophyPhilosophy of Education  and Educational Philosophy
Philosophy of Education and Educational Philosophy
 
Education and training program in the hospital APR.pptx
Education and training program in the hospital APR.pptxEducation and training program in the hospital APR.pptx
Education and training program in the hospital APR.pptx
 
How to Manage Cross-Selling in Odoo 17 Sales
How to Manage Cross-Selling in Odoo 17 SalesHow to Manage Cross-Selling in Odoo 17 Sales
How to Manage Cross-Selling in Odoo 17 Sales
 
Prescribed medication order and communication skills.pptx
Prescribed medication order and communication skills.pptxPrescribed medication order and communication skills.pptx
Prescribed medication order and communication skills.pptx
 
Benefits & Challenges of Inclusive Education
Benefits & Challenges of Inclusive EducationBenefits & Challenges of Inclusive Education
Benefits & Challenges of Inclusive Education
 
Practical Research 1: Lesson 8 Writing the Thesis Statement.pptx
Practical Research 1: Lesson 8 Writing the Thesis Statement.pptxPractical Research 1: Lesson 8 Writing the Thesis Statement.pptx
Practical Research 1: Lesson 8 Writing the Thesis Statement.pptx
 
Prelims of Kant get Marx 2.0: a general politics quiz
Prelims of Kant get Marx 2.0: a general politics quizPrelims of Kant get Marx 2.0: a general politics quiz
Prelims of Kant get Marx 2.0: a general politics quiz
 
How to Add Existing Field in One2Many Tree View in Odoo 17
How to Add Existing Field in One2Many Tree View in Odoo 17How to Add Existing Field in One2Many Tree View in Odoo 17
How to Add Existing Field in One2Many Tree View in Odoo 17
 
How to Solve Singleton Error in the Odoo 17
How to Solve Singleton Error in the  Odoo 17How to Solve Singleton Error in the  Odoo 17
How to Solve Singleton Error in the Odoo 17
 
Easter in the USA presentation by Chloe.
Easter in the USA presentation by Chloe.Easter in the USA presentation by Chloe.
Easter in the USA presentation by Chloe.
 
Patient Counselling. Definition of patient counseling; steps involved in pati...
Patient Counselling. Definition of patient counseling; steps involved in pati...Patient Counselling. Definition of patient counseling; steps involved in pati...
Patient Counselling. Definition of patient counseling; steps involved in pati...
 
UKCGE Parental Leave Discussion March 2024
UKCGE Parental Leave Discussion March 2024UKCGE Parental Leave Discussion March 2024
UKCGE Parental Leave Discussion March 2024
 
Diploma in Nursing Admission Test Question Solution 2023.pdf
Diploma in Nursing Admission Test Question Solution 2023.pdfDiploma in Nursing Admission Test Question Solution 2023.pdf
Diploma in Nursing Admission Test Question Solution 2023.pdf
 
2024.03.23 What do successful readers do - Sandy Millin for PARK.pptx
2024.03.23 What do successful readers do - Sandy Millin for PARK.pptx2024.03.23 What do successful readers do - Sandy Millin for PARK.pptx
2024.03.23 What do successful readers do - Sandy Millin for PARK.pptx
 
Finals of Kant get Marx 2.0 : a general politics quiz
Finals of Kant get Marx 2.0 : a general politics quizFinals of Kant get Marx 2.0 : a general politics quiz
Finals of Kant get Marx 2.0 : a general politics quiz
 

Chapter 3

  • 1. SUBTOPIC 3 : QUANTIFIERS The statement P: n is odd integer. A proposition is a statement that is either true or false. The statement p is not proposition because whether p is true or false depends on the value of n. For example, p is true if n = 104 and false if n = 8. Since, most of the statements in mathematics and computer a science use variable, we must extend the system of logic to include such statements. 1.Quantifiers Definition: Let P (x) be a statement involving the variable x and let D be a set. We call P a proportional function or predicate (with respect to D ) , if for each x ∈ D , P (x) is a proposition. We call D the domain of discourse of P. Example 1: Let P(n) be the statement n is an odd integer Then P is a propositional function with the domain of discourse since for each n ∈ , P(n) is a proposition [for each n ∈ , P(n) is true or false but not both]. For example, if n = 1, we obtain the proposition. P (1): 1 is an odd integer (Which is true) If n = 2, we obtain the proposition/ P (2): 2 is an odd integer (Which is false) 14
  • 2. A propositional function P, by itself, is neither true nor false. However, for each x is domain of discourse, P (x) is a proposition and is, therefore, either true or false. We can think of propositional function as defining a class of propositions, one for each element in the domain of discourse. For example, if P is a propositional function with domain of discourse , we obtain the class of propositions. P (1), P (2), ….. Each of P (1), P (2), …. Is either true or false. 2. Universal Quantification Definition: Let P be a propositional function with the domain of discourse D. The universal quantification of P (x) is the statement. “For all values of x, P is true.” ∀x, P (x) Similar expressions: - For each… - For every… - For any… 3. Counterexample A counterexample is an example chosen to show that a universal statement is FALSE. To verify: - ∀x, P (x) is true - ∀x, P (x) is false 15
  • 3. Example 1: Consider the universally quantified statement. ∀x ( ≥ 0) The domain of discourse is R. The statement is true because for every real number x, it is true that the square of x is positive or zero. According the definition, the universally quantified statement. ∀x, P (x) Is false for at least one x in the domain of discourse that makes P (x) is false. A value x in the domain of discourse that makes P (x) false is called a counterexample to the statement. Example 2: Consider the universally quantified statement. ∀x ( -1 ≥ 0) The domain of discourse is R. The statement is false since, if x = 1, the proposition -1 > 0 Is false. The value1 is counterexample of the statement. ∀x ( -1 ≥ 0) Although there are values of x that make the propositional function true, the counterexample provide show that the universally quantified statement is false. 16
  • 4. 4. Existential Quantification Let P be a proportional function with the domain of discourse D. The existential quantification of P (x) is the statement. “There exists a value of x for which P (x) is true. ∃x, P(x) Similar expressions: - There is some… - There exist… There is at least… Example 1: Consider the existentially quantified statement. ∃x ( The domain of discourse is R. the statement is true because it is possible to find at least one real number x for which the proposition Is true. For example, if x = 2, we obtain the true proposition. 17
  • 5. Is not the case that every value of x results in a true proposition. For example, if x = 1, the proposition Is false. According to definition, the existentially quantified statement ∃x, P(x) Is false for every x in the domain of discourse, the proposition P (x) is false. 18
  • 6. 5. De Morgan’s Law for Logic Theorem: (∀x, P (x)) ≡ (∃x, (P(x)) (∃x, (P(x)) ≡ (∀x, P (x)) The statement “The sum of any two positive real numbers is positive”. ∀x > 0∀y > 0, Example 1: Let P(x) be the statement We show that ∃x, P(x) Is false by verifying that ∀x, ⌐ P (x) Is true. The technique can be justified by appealing to theorem. After we prove that proposition is true, we may negate and conclude that is false. By theorem, ∃x, ⌐⌐P(x) Or equivalently ∃x, P(x) Is also false. 19
  • 7. 1. EXERCISE. a) let P (x) be the propositional function “x ≥ .” Tell whether each proposition is true or false. The domain of discourse is R i. P (1) ii. ⌐∃x P(x) b) Suppose that the domain of discourse of the propositional function P is {1, 2, 3, 4}. Rewrite each propositional function using only negation, disjunction and conjunction. i. ∀x P (x) c) Determine the truth value, the domain discourse R x R, justify the answer. i. ∀x ∀y ( < y + 1) d) Assume that ∀x ∃y P(x, y) is false and that the domain of discourse is nonempty. It must also be false? Prove the answer. i. ∀x ∀y P (x,y) 20