SlideShare una empresa de Scribd logo
1 de 14
Chapter 1- Preliminaries
1.1 History of Algebra
The word “algebra”-al jebr (in Arabic)
• was first used by Mohammed Al-Khwarizmi -muslim Math.
• ninth century, when taught mathematics in Baghdad.
• means “reunion”,decribes his method for collecting the terms
of an equations in order to solve it.
• Omar Khayyam, another Mathematician, defined it as
the science of solving equations.

Elementary Algebra (Classical age of Algebra)
• its central theme is clearly identified as the solving of eqs.
- method of solving linear, qudratic, cubic, quartic equations.
- 1824, Niels Abel – there does not exits any formula for
1
equations degree 5 or greater.
Modern Age

• new varieties of algebra arose
connection with the application in math to practical problems.
- Matrix Algebra
- Bolean Algebra
- Algebra of vectors and tensors
- ~200 different kinds of algebra.
• the awareness grew
- algebra can no longer be conceived merely as the
science of solving equations.
- It had to be viewed as much more broadly as a
branch of mathematics.
revealing general principles which apply equally to
all known and all possible algebras:
* What is it that all algebras have in common?
* What trait do they share which lets us refer to all of
them as algebras? Algebraic Structure
• Abstract Algebra (Modern Algebra) -more adv. course
2
The study algebraic structures.
1.2 Logic and Proof
Undefined Terms
- Understand these terms and feel comfortable using them
to define new terms.

Important Terms
• Statement or Proposition
- Declarative sentence that is either true or false, but not both.
• Postulates
- Statements that are assumed to be true.
• Definition
- A precise meaning to a mathematical term.
3
• Theorem
- A major landmark in the mathematical theory.
- Postulates and definitions are used to prove theorems.
- Once a theorem is proved to be true, it can be used.
• Lemma
- A result that is needed to prove a theorem.
• Corollary
- A result that follows immediately from a theorem.
• Example
- Is not a general result but is a particular case.
• Proof
- Mathematical argument intended to convince us that
a result is correct.

4
Conjunction, Disjunction and Negation
Definition:
Let P and Q be statements.
i) The statement P AND Q, P ∧ Q,
is called the conjunction of P and Q.
ii) The statement P OR Q, P ∨ Q,
is called the disjunction of P and Q.
iii) The negation of P is denoted by NOT P or ~ P
Conditional and Biconditional Statement
Conditional statement: “If P then Q”, P ⇒ Q.
Biconditional statement: “P if and only if Q”, P ⇔ Q.
5
Quantifiers
Consider a statement P(x) :

x >5

- Statement P(x) is depending on the variable x.
- Adding quantifiers can convert statement P(x) into a
statement that is either true or false.
• Universal quantifier

(∀)

P(x) is true for all values of x, denoted by

∀x, P ( x)

or
For all x, P(x).
For every x, P(x).
For each x, P(x).
P(x), for all x.

6
• Existential quantifier (∃)
There exist an x for which P(x) is true
or
For some x, P(x).
P(x), for some x.

: ∃x, P ( x)

Example
1.

∀x ∈ R, x −1 = ( x −1)( x + x + 1)
3

2

- True or false statement? Why?
2.

∀x ∈ R, x + x − 6 = 0
2

- True or False statement? Why?
3.

∃x ∈ R, x + x − 6 = 0
2

- True or False statement? Why?

7
Proofs
- Many mathematical theorems can be expressed
symbolically in the form of

P ⇒Q
Assumption
Or
hypothesis

Conclusion

may consists of one or more
statements.

- The theorem says that if the assumption is true than the
conclusion is true.
- How do you go about thinking up ways to prove a
theorem?
• Understand the definitions
• Try examples
• Try standard proof methods

8
Methods of Proof ( P ⇒ Q)
1. Direct Method
• find a series of statements P1,P2,…,Pn
• verify that each of the implications below is true
P →P , P →P2 , P2 →P3 .....Pn −1 →Pn and Pn → Q
1
1
Example
An integer n is defined to be even if n = 2m for some integer m.
Show that the sum of two even integers is even.
Proof

9
2. Contrapositive Method
• may prove ¬Q → ¬P
Example
Proposition:
If x is a real number such that

x + 7 x < 9, then x < 1.1
3

2

Proof

10
3. Proof by Contradiction
• assume that P is true and not Q is true (Q is false)
• will end up with a false statement S
• Conclude that not Q must be false, i.e., Q is true
Example
Proposition:
If x is an integer and x2 is even then x is an even integer.
Proof

11
4. Proof by Induction
• assume that for each positive integer n,
a statement P(n) is given. If
1. P(1) is a true statement; and
2. Whenever P(k) is a true statement, then P(k+1) is also true,
• then P(n) is a true statement for every n in positive integer.
Example
Prove:

1
1
1
1
n
+
+
+ ... +
=
1• 3 3 • 5 5 • 7
(2n − 1)(2n + 1) 2n + 1
Proof
12
5. Proof by Counterexamples
• Sometimes a conjectured result in mathematics is not true.
• Would not be able to prove it.
• Could try to disprove it.
• The conjecture in the form of ∀x, P ( x )
• Take the negation: NOT (∀x, P ( x ))
Equivalent to:

∃x, NOT P ( x)

• Hence to disprove the statement

∀x, P ( x)

need only to find one value, say c, such that P(c) is false.
• The value c is called a counterexample to the conjecture.

13
Example
Let x be a real number. Disprove the statement
If x2 >9 then x >3.
Solution

Remark
• To disprove the conjecture in the form of

∃x, P ( x)

cannot use counter example!!!
Its negation is equivalently in the form of ∀x, NOT
Need to show that P(x) is false for all values of x.
• To prove

P ( x)

P ⇔ Q : Prove P ⇒Q and Q ⇒P

14

Más contenido relacionado

La actualidad más candente

Disrete mathematics and_its application_by_rosen _7th edition_lecture_1
Disrete mathematics and_its application_by_rosen _7th edition_lecture_1Disrete mathematics and_its application_by_rosen _7th edition_lecture_1
Disrete mathematics and_its application_by_rosen _7th edition_lecture_1
taimoor iftikhar
 
Goldbach Conjecture
Goldbach ConjectureGoldbach Conjecture
Goldbach Conjecture
Anil1091
 
1.1 patterns & inductive reasoning
1.1 patterns & inductive reasoning1.1 patterns & inductive reasoning
1.1 patterns & inductive reasoning
vickihoover
 
5.2 arithmetic sequences and sums
5.2 arithmetic sequences and sums5.2 arithmetic sequences and sums
5.2 arithmetic sequences and sums
math260
 
Geometry 1.1 patterns and inductive reasoning
Geometry 1.1 patterns and inductive reasoningGeometry 1.1 patterns and inductive reasoning
Geometry 1.1 patterns and inductive reasoning
kca1528
 
Ch1 sets and_logic(1)
Ch1 sets and_logic(1)Ch1 sets and_logic(1)
Ch1 sets and_logic(1)
Kwonpyo Ko
 

La actualidad más candente (19)

Inductive Reasoning
Inductive ReasoningInductive Reasoning
Inductive Reasoning
 
Ppt on polynomial
Ppt on polynomial Ppt on polynomial
Ppt on polynomial
 
Deductive and Inductive Reasoning with Vizzini
Deductive and Inductive Reasoning with VizziniDeductive and Inductive Reasoning with Vizzini
Deductive and Inductive Reasoning with Vizzini
 
Polynomials Class 9th
Polynomials Class 9thPolynomials Class 9th
Polynomials Class 9th
 
Disrete mathematics and_its application_by_rosen _7th edition_lecture_1
Disrete mathematics and_its application_by_rosen _7th edition_lecture_1Disrete mathematics and_its application_by_rosen _7th edition_lecture_1
Disrete mathematics and_its application_by_rosen _7th edition_lecture_1
 
Remainder and Factor Theorem
Remainder and Factor TheoremRemainder and Factor Theorem
Remainder and Factor Theorem
 
Goldbach Conjecture
Goldbach ConjectureGoldbach Conjecture
Goldbach Conjecture
 
Logic
LogicLogic
Logic
 
Math 150 fall 2020 homework 1 due date friday, october 15,
Math 150 fall 2020 homework 1 due date friday, october 15,Math 150 fall 2020 homework 1 due date friday, october 15,
Math 150 fall 2020 homework 1 due date friday, october 15,
 
1.1 patterns & inductive reasoning
1.1 patterns & inductive reasoning1.1 patterns & inductive reasoning
1.1 patterns & inductive reasoning
 
Applications of set theory
Applications of  set theoryApplications of  set theory
Applications of set theory
 
BCA_Semester-I_Mathematics-I_Set theory and function
BCA_Semester-I_Mathematics-I_Set theory and functionBCA_Semester-I_Mathematics-I_Set theory and function
BCA_Semester-I_Mathematics-I_Set theory and function
 
On Review of the Cluster Point of a Set in a Topological Space
On Review of the Cluster Point of a Set in a Topological SpaceOn Review of the Cluster Point of a Set in a Topological Space
On Review of the Cluster Point of a Set in a Topological Space
 
5.2 arithmetic sequences and sums
5.2 arithmetic sequences and sums5.2 arithmetic sequences and sums
5.2 arithmetic sequences and sums
 
Set theory
Set theorySet theory
Set theory
 
Geometry 1.1 patterns and inductive reasoning
Geometry 1.1 patterns and inductive reasoningGeometry 1.1 patterns and inductive reasoning
Geometry 1.1 patterns and inductive reasoning
 
Set Theory
Set TheorySet Theory
Set Theory
 
Ch1 sets and_logic(1)
Ch1 sets and_logic(1)Ch1 sets and_logic(1)
Ch1 sets and_logic(1)
 
1.3.1B Inductive Reasoning
1.3.1B Inductive Reasoning1.3.1B Inductive Reasoning
1.3.1B Inductive Reasoning
 

Destacado

Destacado (11)

MATH GRADE 10 LEARNER'S MODULE
MATH GRADE 10 LEARNER'S MODULEMATH GRADE 10 LEARNER'S MODULE
MATH GRADE 10 LEARNER'S MODULE
 
What Makes Great Infographics
What Makes Great InfographicsWhat Makes Great Infographics
What Makes Great Infographics
 
Masters of SlideShare
Masters of SlideShareMasters of SlideShare
Masters of SlideShare
 
STOP! VIEW THIS! 10-Step Checklist When Uploading to Slideshare
STOP! VIEW THIS! 10-Step Checklist When Uploading to SlideshareSTOP! VIEW THIS! 10-Step Checklist When Uploading to Slideshare
STOP! VIEW THIS! 10-Step Checklist When Uploading to Slideshare
 
You Suck At PowerPoint!
You Suck At PowerPoint!You Suck At PowerPoint!
You Suck At PowerPoint!
 
10 Ways to Win at SlideShare SEO & Presentation Optimization
10 Ways to Win at SlideShare SEO & Presentation Optimization10 Ways to Win at SlideShare SEO & Presentation Optimization
10 Ways to Win at SlideShare SEO & Presentation Optimization
 
How To Get More From SlideShare - Super-Simple Tips For Content Marketing
How To Get More From SlideShare - Super-Simple Tips For Content MarketingHow To Get More From SlideShare - Super-Simple Tips For Content Marketing
How To Get More From SlideShare - Super-Simple Tips For Content Marketing
 
2015 Upload Campaigns Calendar - SlideShare
2015 Upload Campaigns Calendar - SlideShare2015 Upload Campaigns Calendar - SlideShare
2015 Upload Campaigns Calendar - SlideShare
 
What to Upload to SlideShare
What to Upload to SlideShareWhat to Upload to SlideShare
What to Upload to SlideShare
 
How to Make Awesome SlideShares: Tips & Tricks
How to Make Awesome SlideShares: Tips & TricksHow to Make Awesome SlideShares: Tips & Tricks
How to Make Awesome SlideShares: Tips & Tricks
 
Getting Started With SlideShare
Getting Started With SlideShareGetting Started With SlideShare
Getting Started With SlideShare
 

Similar a Tma2033 chap1.1&1.2handout

Mathematical reasoning
Mathematical reasoningMathematical reasoning
Mathematical reasoning
Aza Alias
 
Mcs lecture19.methods ofproof(1)
Mcs lecture19.methods ofproof(1)Mcs lecture19.methods ofproof(1)
Mcs lecture19.methods ofproof(1)
kevinwu1994
 

Similar a Tma2033 chap1.1&1.2handout (20)

MATHEMATICAL INDUCTION.ppt
MATHEMATICAL INDUCTION.pptMATHEMATICAL INDUCTION.ppt
MATHEMATICAL INDUCTION.ppt
 
Logic
LogicLogic
Logic
 
Presentation1.pptx
Presentation1.pptxPresentation1.pptx
Presentation1.pptx
 
Discrete Structure Lecture #5 & 6.pdf
Discrete Structure Lecture #5 & 6.pdfDiscrete Structure Lecture #5 & 6.pdf
Discrete Structure Lecture #5 & 6.pdf
 
Mathematical reasoning
Mathematical reasoningMathematical reasoning
Mathematical reasoning
 
Mcs lecture19.methods ofproof(1)
Mcs lecture19.methods ofproof(1)Mcs lecture19.methods ofproof(1)
Mcs lecture19.methods ofproof(1)
 
Method of direct proof
Method of direct proofMethod of direct proof
Method of direct proof
 
desmath(1).ppt
desmath(1).pptdesmath(1).ppt
desmath(1).ppt
 
Predicates and Quantifiers
Predicates and Quantifiers Predicates and Quantifiers
Predicates and Quantifiers
 
Predicates and quantifiers
Predicates and quantifiersPredicates and quantifiers
Predicates and quantifiers
 
Logic.ppt
Logic.pptLogic.ppt
Logic.ppt
 
CMSC 56 | Lecture 5: Proofs Methods and Strategy
CMSC 56 | Lecture 5: Proofs Methods and StrategyCMSC 56 | Lecture 5: Proofs Methods and Strategy
CMSC 56 | Lecture 5: Proofs Methods and Strategy
 
PropositionalLogic.ppt
PropositionalLogic.pptPropositionalLogic.ppt
PropositionalLogic.ppt
 
Course notes1
Course notes1Course notes1
Course notes1
 
Logic agent
Logic agentLogic agent
Logic agent
 
Discrete mathematics suraj ppt
Discrete mathematics suraj pptDiscrete mathematics suraj ppt
Discrete mathematics suraj ppt
 
Theory of Knowledge - mathematics philosophies
Theory of Knowledge -  mathematics philosophiesTheory of Knowledge -  mathematics philosophies
Theory of Knowledge - mathematics philosophies
 
AI-Unit4.ppt
AI-Unit4.pptAI-Unit4.ppt
AI-Unit4.ppt
 
1019Lec1.ppt
1019Lec1.ppt1019Lec1.ppt
1019Lec1.ppt
 
Predicate logic_2(Artificial Intelligence)
Predicate logic_2(Artificial Intelligence)Predicate logic_2(Artificial Intelligence)
Predicate logic_2(Artificial Intelligence)
 

Último

Spellings Wk 3 English CAPS CARES Please Practise
Spellings Wk 3 English CAPS CARES Please PractiseSpellings Wk 3 English CAPS CARES Please Practise
Spellings Wk 3 English CAPS CARES Please Practise
AnaAcapella
 
Seal of Good Local Governance (SGLG) 2024Final.pptx
Seal of Good Local Governance (SGLG) 2024Final.pptxSeal of Good Local Governance (SGLG) 2024Final.pptx
Seal of Good Local Governance (SGLG) 2024Final.pptx
negromaestrong
 
Jual Obat Aborsi Hongkong ( Asli No.1 ) 085657271886 Obat Penggugur Kandungan...
Jual Obat Aborsi Hongkong ( Asli No.1 ) 085657271886 Obat Penggugur Kandungan...Jual Obat Aborsi Hongkong ( Asli No.1 ) 085657271886 Obat Penggugur Kandungan...
Jual Obat Aborsi Hongkong ( Asli No.1 ) 085657271886 Obat Penggugur Kandungan...
ZurliaSoop
 
1029 - Danh muc Sach Giao Khoa 10 . pdf
1029 -  Danh muc Sach Giao Khoa 10 . pdf1029 -  Danh muc Sach Giao Khoa 10 . pdf
1029 - Danh muc Sach Giao Khoa 10 . pdf
QucHHunhnh
 
Russian Escort Service in Delhi 11k Hotel Foreigner Russian Call Girls in Delhi
Russian Escort Service in Delhi 11k Hotel Foreigner Russian Call Girls in DelhiRussian Escort Service in Delhi 11k Hotel Foreigner Russian Call Girls in Delhi
Russian Escort Service in Delhi 11k Hotel Foreigner Russian Call Girls in Delhi
kauryashika82
 
1029-Danh muc Sach Giao Khoa khoi 6.pdf
1029-Danh muc Sach Giao Khoa khoi  6.pdf1029-Danh muc Sach Giao Khoa khoi  6.pdf
1029-Danh muc Sach Giao Khoa khoi 6.pdf
QucHHunhnh
 

Último (20)

Making communications land - Are they received and understood as intended? we...
Making communications land - Are they received and understood as intended? we...Making communications land - Are they received and understood as intended? we...
Making communications land - Are they received and understood as intended? we...
 
Spellings Wk 3 English CAPS CARES Please Practise
Spellings Wk 3 English CAPS CARES Please PractiseSpellings Wk 3 English CAPS CARES Please Practise
Spellings Wk 3 English CAPS CARES Please Practise
 
Seal of Good Local Governance (SGLG) 2024Final.pptx
Seal of Good Local Governance (SGLG) 2024Final.pptxSeal of Good Local Governance (SGLG) 2024Final.pptx
Seal of Good Local Governance (SGLG) 2024Final.pptx
 
psychiatric nursing HISTORY COLLECTION .docx
psychiatric  nursing HISTORY  COLLECTION  .docxpsychiatric  nursing HISTORY  COLLECTION  .docx
psychiatric nursing HISTORY COLLECTION .docx
 
PROCESS RECORDING FORMAT.docx
PROCESS      RECORDING        FORMAT.docxPROCESS      RECORDING        FORMAT.docx
PROCESS RECORDING FORMAT.docx
 
Jual Obat Aborsi Hongkong ( Asli No.1 ) 085657271886 Obat Penggugur Kandungan...
Jual Obat Aborsi Hongkong ( Asli No.1 ) 085657271886 Obat Penggugur Kandungan...Jual Obat Aborsi Hongkong ( Asli No.1 ) 085657271886 Obat Penggugur Kandungan...
Jual Obat Aborsi Hongkong ( Asli No.1 ) 085657271886 Obat Penggugur Kandungan...
 
2024-NATIONAL-LEARNING-CAMP-AND-OTHER.pptx
2024-NATIONAL-LEARNING-CAMP-AND-OTHER.pptx2024-NATIONAL-LEARNING-CAMP-AND-OTHER.pptx
2024-NATIONAL-LEARNING-CAMP-AND-OTHER.pptx
 
Application orientated numerical on hev.ppt
Application orientated numerical on hev.pptApplication orientated numerical on hev.ppt
Application orientated numerical on hev.ppt
 
1029 - Danh muc Sach Giao Khoa 10 . pdf
1029 -  Danh muc Sach Giao Khoa 10 . pdf1029 -  Danh muc Sach Giao Khoa 10 . pdf
1029 - Danh muc Sach Giao Khoa 10 . pdf
 
Sociology 101 Demonstration of Learning Exhibit
Sociology 101 Demonstration of Learning ExhibitSociology 101 Demonstration of Learning Exhibit
Sociology 101 Demonstration of Learning Exhibit
 
ICT role in 21st century education and it's challenges.
ICT role in 21st century education and it's challenges.ICT role in 21st century education and it's challenges.
ICT role in 21st century education and it's challenges.
 
Asian American Pacific Islander Month DDSD 2024.pptx
Asian American Pacific Islander Month DDSD 2024.pptxAsian American Pacific Islander Month DDSD 2024.pptx
Asian American Pacific Islander Month DDSD 2024.pptx
 
Python Notes for mca i year students osmania university.docx
Python Notes for mca i year students osmania university.docxPython Notes for mca i year students osmania university.docx
Python Notes for mca i year students osmania university.docx
 
Holdier Curriculum Vitae (April 2024).pdf
Holdier Curriculum Vitae (April 2024).pdfHoldier Curriculum Vitae (April 2024).pdf
Holdier Curriculum Vitae (April 2024).pdf
 
ComPTIA Overview | Comptia Security+ Book SY0-701
ComPTIA Overview | Comptia Security+ Book SY0-701ComPTIA Overview | Comptia Security+ Book SY0-701
ComPTIA Overview | Comptia Security+ Book SY0-701
 
Mixin Classes in Odoo 17 How to Extend Models Using Mixin Classes
Mixin Classes in Odoo 17  How to Extend Models Using Mixin ClassesMixin Classes in Odoo 17  How to Extend Models Using Mixin Classes
Mixin Classes in Odoo 17 How to Extend Models Using Mixin Classes
 
Explore beautiful and ugly buildings. Mathematics helps us create beautiful d...
Explore beautiful and ugly buildings. Mathematics helps us create beautiful d...Explore beautiful and ugly buildings. Mathematics helps us create beautiful d...
Explore beautiful and ugly buildings. Mathematics helps us create beautiful d...
 
Russian Escort Service in Delhi 11k Hotel Foreigner Russian Call Girls in Delhi
Russian Escort Service in Delhi 11k Hotel Foreigner Russian Call Girls in DelhiRussian Escort Service in Delhi 11k Hotel Foreigner Russian Call Girls in Delhi
Russian Escort Service in Delhi 11k Hotel Foreigner Russian Call Girls in Delhi
 
TỔNG ÔN TẬP THI VÀO LỚP 10 MÔN TIẾNG ANH NĂM HỌC 2023 - 2024 CÓ ĐÁP ÁN (NGỮ Â...
TỔNG ÔN TẬP THI VÀO LỚP 10 MÔN TIẾNG ANH NĂM HỌC 2023 - 2024 CÓ ĐÁP ÁN (NGỮ Â...TỔNG ÔN TẬP THI VÀO LỚP 10 MÔN TIẾNG ANH NĂM HỌC 2023 - 2024 CÓ ĐÁP ÁN (NGỮ Â...
TỔNG ÔN TẬP THI VÀO LỚP 10 MÔN TIẾNG ANH NĂM HỌC 2023 - 2024 CÓ ĐÁP ÁN (NGỮ Â...
 
1029-Danh muc Sach Giao Khoa khoi 6.pdf
1029-Danh muc Sach Giao Khoa khoi  6.pdf1029-Danh muc Sach Giao Khoa khoi  6.pdf
1029-Danh muc Sach Giao Khoa khoi 6.pdf
 

Tma2033 chap1.1&1.2handout

  • 1. Chapter 1- Preliminaries 1.1 History of Algebra The word “algebra”-al jebr (in Arabic) • was first used by Mohammed Al-Khwarizmi -muslim Math. • ninth century, when taught mathematics in Baghdad. • means “reunion”,decribes his method for collecting the terms of an equations in order to solve it. • Omar Khayyam, another Mathematician, defined it as the science of solving equations. Elementary Algebra (Classical age of Algebra) • its central theme is clearly identified as the solving of eqs. - method of solving linear, qudratic, cubic, quartic equations. - 1824, Niels Abel – there does not exits any formula for 1 equations degree 5 or greater.
  • 2. Modern Age • new varieties of algebra arose connection with the application in math to practical problems. - Matrix Algebra - Bolean Algebra - Algebra of vectors and tensors - ~200 different kinds of algebra. • the awareness grew - algebra can no longer be conceived merely as the science of solving equations. - It had to be viewed as much more broadly as a branch of mathematics. revealing general principles which apply equally to all known and all possible algebras: * What is it that all algebras have in common? * What trait do they share which lets us refer to all of them as algebras? Algebraic Structure • Abstract Algebra (Modern Algebra) -more adv. course 2 The study algebraic structures.
  • 3. 1.2 Logic and Proof Undefined Terms - Understand these terms and feel comfortable using them to define new terms. Important Terms • Statement or Proposition - Declarative sentence that is either true or false, but not both. • Postulates - Statements that are assumed to be true. • Definition - A precise meaning to a mathematical term. 3
  • 4. • Theorem - A major landmark in the mathematical theory. - Postulates and definitions are used to prove theorems. - Once a theorem is proved to be true, it can be used. • Lemma - A result that is needed to prove a theorem. • Corollary - A result that follows immediately from a theorem. • Example - Is not a general result but is a particular case. • Proof - Mathematical argument intended to convince us that a result is correct. 4
  • 5. Conjunction, Disjunction and Negation Definition: Let P and Q be statements. i) The statement P AND Q, P ∧ Q, is called the conjunction of P and Q. ii) The statement P OR Q, P ∨ Q, is called the disjunction of P and Q. iii) The negation of P is denoted by NOT P or ~ P Conditional and Biconditional Statement Conditional statement: “If P then Q”, P ⇒ Q. Biconditional statement: “P if and only if Q”, P ⇔ Q. 5
  • 6. Quantifiers Consider a statement P(x) : x >5 - Statement P(x) is depending on the variable x. - Adding quantifiers can convert statement P(x) into a statement that is either true or false. • Universal quantifier (∀) P(x) is true for all values of x, denoted by ∀x, P ( x) or For all x, P(x). For every x, P(x). For each x, P(x). P(x), for all x. 6
  • 7. • Existential quantifier (∃) There exist an x for which P(x) is true or For some x, P(x). P(x), for some x. : ∃x, P ( x) Example 1. ∀x ∈ R, x −1 = ( x −1)( x + x + 1) 3 2 - True or false statement? Why? 2. ∀x ∈ R, x + x − 6 = 0 2 - True or False statement? Why? 3. ∃x ∈ R, x + x − 6 = 0 2 - True or False statement? Why? 7
  • 8. Proofs - Many mathematical theorems can be expressed symbolically in the form of P ⇒Q Assumption Or hypothesis Conclusion may consists of one or more statements. - The theorem says that if the assumption is true than the conclusion is true. - How do you go about thinking up ways to prove a theorem? • Understand the definitions • Try examples • Try standard proof methods 8
  • 9. Methods of Proof ( P ⇒ Q) 1. Direct Method • find a series of statements P1,P2,…,Pn • verify that each of the implications below is true P →P , P →P2 , P2 →P3 .....Pn −1 →Pn and Pn → Q 1 1 Example An integer n is defined to be even if n = 2m for some integer m. Show that the sum of two even integers is even. Proof 9
  • 10. 2. Contrapositive Method • may prove ¬Q → ¬P Example Proposition: If x is a real number such that x + 7 x < 9, then x < 1.1 3 2 Proof 10
  • 11. 3. Proof by Contradiction • assume that P is true and not Q is true (Q is false) • will end up with a false statement S • Conclude that not Q must be false, i.e., Q is true Example Proposition: If x is an integer and x2 is even then x is an even integer. Proof 11
  • 12. 4. Proof by Induction • assume that for each positive integer n, a statement P(n) is given. If 1. P(1) is a true statement; and 2. Whenever P(k) is a true statement, then P(k+1) is also true, • then P(n) is a true statement for every n in positive integer. Example Prove: 1 1 1 1 n + + + ... + = 1• 3 3 • 5 5 • 7 (2n − 1)(2n + 1) 2n + 1 Proof 12
  • 13. 5. Proof by Counterexamples • Sometimes a conjectured result in mathematics is not true. • Would not be able to prove it. • Could try to disprove it. • The conjecture in the form of ∀x, P ( x ) • Take the negation: NOT (∀x, P ( x )) Equivalent to: ∃x, NOT P ( x) • Hence to disprove the statement ∀x, P ( x) need only to find one value, say c, such that P(c) is false. • The value c is called a counterexample to the conjecture. 13
  • 14. Example Let x be a real number. Disprove the statement If x2 >9 then x >3. Solution Remark • To disprove the conjecture in the form of ∃x, P ( x) cannot use counter example!!! Its negation is equivalently in the form of ∀x, NOT Need to show that P(x) is false for all values of x. • To prove P ( x) P ⇔ Q : Prove P ⇒Q and Q ⇒P 14