Mathematics I

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Mathematics I - Introduction

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Why study Math?

http://www.karlin.mff.cuni.cz/∼pick/2018-10-02-prvni-prednaska-z-analyzy.pdf

https://www.youtube.com/watch?v=6eC3ndnR86s https://fanmovie.cz/dvd/alenka-v-risi-divu-na-dvd-a-blu-ray/ http://ceskapozice.lidovky.cz/nove-relikvie-z-mendelovy-pozustalosti-otec-genetiky-laka-vedce-do-ceska-1q6- /tema.aspx?c=A121220 120247 pozice 88014

https://g.cz/pred-77-lety-byl-cepinem-zavrazden-bolsevik-trockij-po-propusteni-z-vezeni-zil-jeho-vrah-kousek-za-prahou/

Mathematics I - Introduction 2 / 48

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Why study Math?

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Why study Math?

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Why study Math?

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Why study Math?

https://www.youtube.com/watch?v=6eC3ndnR86s https://fanmovie.cz/dvd/alenka-v-risi-divu-na-dvd-a-blu-ray/

http://ceskapozice.lidovky.cz/nove-relikvie-z-mendelovy-pozustalosti-otec-genetiky-laka-vedce-do-ceska-1q6- /tema.aspx?c=A121220 120247 pozice 88014

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Goal of the course

Preparation for other courses — Statistics, Microeconomics, . . .

Training of logical thinking and mathematical exactness

At the end of the course students should be able to

compute limits and derivatives and investigate functions understand definitions (give positive and negative

examples) and theorems (explain their meaning, neccessity of the assumptions, apply them in particular situations) perform mathematical proofs, give mathematically exact arguments, write mathematical formulae, use quantifiers

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Goal of the course

Training of logical thinking and mathematical exactness At the end of the course students should be able to

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Goal of the course

compute limits and derivatives and investigate functions

understand definitions (give positive and negative

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Goal of the course

examples) and theorems (explain their meaning, neccessity of the assumptions, apply them in particular situations)

perform mathematical proofs, give mathematically exact arguments, write mathematical formulae, use quantifiers

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Goal of the course

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Introduction

Limit of a sequence Mappings

Functions of one real variable

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Textbooks

H´ajkov´a et al: Mathematics 1 Trench: Introduction to real analysis

Ghorpade, Limaye: A course in calculus and real analysis Zorich: Mathematical analysis I

Rudin: Principles of mathematical analysis

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Sets

We take a set to be a collection of definite and distinguishable objects into a coherent whole.

x∈A. . .xis an element (or member) of the setA

Exercise (True or false)

A- set of all animals living in Australia.

A a∈A B b∈A C c∈A D d ∈A E e∈A

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Sets

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Sets

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Sets

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Sets

x∈A. . .xis an element (or member) of the setA Exercise (True or false)

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Sets

x∈/ A. . .xis not a member of the setA

A a̸∈A B b̸∈A C c̸∈A D d ̸∈A E e̸∈A

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Sets

x∈/ A. . .xis not a member of the setA Exercise (True or false)

A a̸∈A B b̸∈A C c̸∈A D d ̸∈A E e̸∈A

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Sets

A^c. . . the complement of the setA

B⊂A. . . the setBis a subset of the setA(inclusion) Example:Bis the set of all birds living in Australia:B⊂A. A=B. . . the setsAandBhave the same elements; the following holds:A⊂BandB⊂A

∅. . . an empty set

A∪B. . . the union of the setsAandB A∩B. . . the intersection of the setsAandB disjoint sets . . .AandBare disjoint ifA∩B=∅

A\B={x∈A; x∈/ B}. . . a difference of the setsAandB A₁× · · · ×Am={[a₁, . . . ,am]; a₁ ∈A₁, . . . ,am∈Am} . . . the Cartesian product

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Sets

B⊂A. . . the setBis a subset of the setA(inclusion) Example:Bis the set of all birds living in Australia:B⊂A.

A=B. . . the setsAandBhave the same elements; the following holds:A⊂BandB⊂A

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Sets

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Sets

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Sets

A∪B. . . the union of the setsAandB

A∩B. . . the intersection of the setsAandB disjoint sets . . .AandBare disjoint ifA∩B=∅

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Sets

A∪B. . . the union of the setsAandB A∩B. . . the intersection of the setsAandB

disjoint sets . . .AandBare disjoint ifA∩B=∅

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Sets

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Sets

A\B={x∈A; x∈/ B}. . . a difference of the setsAandB

A₁× · · · ×Am={[a₁, . . . ,am]; a₁ ∈A₁, . . . ,am∈Am} . . . the Cartesian product

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Sets

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Sets - questions

Exercise

LetU ={1,2,3,4,5,6,7,8,9},A={1,3,5,7,9}and B={1,2,3,4,5}. Find

1. A∪B 2. A∩B

3. A^c 4. (B^c)^c

5. A\B 6. B\A

Exercise (True or false) LetAbe a set.

A ∅ ∈A B ∅ ⊂A C 0=∅

D {x} ∈ {x,y,z} E x∈ {x,y,z}

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Sets - questions

Exercise

LetU ={1,2,3,4,5,6,7,8,9},A={1,3,5,7,9}and B={1,2,3,4,5}. Find

1. A∪B 2. A∩B

3. A^c 4. (B^c)^c

5. A\B 6. B\A

Exercise (True or false) LetAbe a set.

A ∅ ∈A B ∅ ⊂A C 0=∅

D {x} ∈ {x,y,z}

E x∈ {x,y,z}

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Sets

A₁× · · · ×A_m={[a₁, . . . ,a_m]; a₁ ∈A₁, . . . ,a_m∈A_m}. . . the Cartesian product

Exercise

LetA={1,2,3},B={2,4}. FindA×B,B×Band sketch them.

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Sets

A₁× · · · ×A_m={[a₁, . . . ,a_m]; a₁ ∈A₁, . . . ,a_m∈A_m}. . . the Cartesian product

Exercise

LetA={1,2,3},B={2,4}. FindA×B,B×Band sketch them.

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Sets

LetIbe a non-empty set of indices and suppose we have a system of setsA_α, where the indicesαrun overI.

S

α∈I

A_α. . . the set of all elements belonging to at least one of the setsA_α

T

α∈I

A_α. . . the set of all elements belonging to everyA_α Example.

A₁∪A₂∪A₃is equivalent to

3

S

i=1

A_i, and also to S

i∈{1,2,3}

A_i. Infinitely many sets:A₁∪A₂∪A₃∪. . . is equivalent to

∞

S

i=1

Ai, and also to S

i∈N

A_i.

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Sets

S

α∈I

T

α∈I

A_α. . . the set of all elements belonging to everyA_α

Example.

3

S

i=1

A_i, and also to S

i∈{1,2,3}

∞

S

i=1

Ai, and also to S

i∈N

A_i.

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Sets

S

α∈I

T

α∈I

A_α. . . the set of all elements belonging to everyA_α

Example.

3

S

i=1

A_i, and also to S

i∈{1,2,3}

∞

S

i=1

Ai, and also to S

i∈N

A_i.

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Sets

S

α∈I

T

α∈I

3

S

i=1

A_i, and also to S

i∈{1,2,3}

∞

S

i=1

Ai, and also to S

i∈N

A_i.

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Sets

S

α∈I

T

α∈I

3

S

i=1

A_i, and also to S

i∈{1,2,3}

A_i.

Infinitely many sets:A₁∪A₂∪A₃∪. . . is equivalent to

∞

S

i=1

Ai, and also to S

i∈N

A_i.

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Sets

S

α∈I

T

α∈I

3

S

i=1

A_i, and also to S

i∈{1,2,3}

∞

S

i=1

Ai, and also to S

i∈N

A_i.

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Sets

Exercise

LetA₁ ={0,1},A₂ ={0,2},A₃ ={0,3}. Find 1.

3

[

i=1

A_i 2. \

i∈{1,2,3}

Ai

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Logic

Astatement(or proposition) is a sentence which can be declared to be either true or false.

Exercise

Find statements. A Let it be!

B We all live in a yellow submarine. C Is there anybody out there? D We don’t need no education.

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Logic

Exercise

Find statements. A Let it be!

B We all live in a yellow submarine. C Is there anybody out there? D We don’t need no education.

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Logic

Exercise

Find statements.

A Let it be!

B We all live in a yellow submarine.

C Is there anybody out there?

D We don’t need no education.

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Statements

¬, alsonon. . .negation

& (also∧) . . .conjunction, logical “and”

∨. . .disjuction(alternative), logical “or”

⇒. . .implication

⇔. . .equivalence; “if and only if”

Exercise

1. Alice does not like chocolate icecream.

2. Alice likes chocolate and lemon icecream.

3. Alice likes chocolate or lemon icecream.

4. If it will be raining tomorrow, we will play board games.

5. We will play board games tomorrow if and only if it will be raining.

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Apredicate(or propositional function) is an expression or sentence involving variables which becomes a statement once we substitute certain elements of a given set for the variables.

General form:

V(x),x∈M

V(x₁, . . . ,x_n),x₁ ∈M₁, . . . ,x_n ∈M_n

Example V(x):xis even M={1,2,3,4,5} V(3)false,V(4)true. V(x₁,x₂):x₁·x₂ =1 M={2,¹₂,3,4}

V(2,¹₂)true,V(2,3)false.

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General form:

V(x),x∈M

V(x₁, . . . ,x_n),x₁ ∈M₁, . . . ,x_n ∈M_n

Example V(x):xis even M={1,2,3,4,5} V(3)false,V(4)true. V(x₁,x₂):x₁·x₂ =1 M={2,¹₂,3,4}

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General form:

V(x),x∈M

V(x₁, . . . ,x_n),x₁ ∈M₁, . . . ,x_n ∈M_n

Example V(x):xis even M={1,2,3,4,5}

V(3)false,V(4)true.

V(x₁,x₂):x₁·x₂ =1 M={2,¹₂,3,4}

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IfA(x),x∈M is a predicate, then the statement “A(x)holds for everyxfromM.” is shortened to

∀x∈M: A(x).

The statement “There existsxinM such thatA(x)holds.” is shortened to

∃x∈M: A(x).

The statement “There is only onexinMsuch thatA(x)holds.” is shortened to

∃!x∈M: A(x).

Example

∀x∈R:|x| ≥0

∃x∈Q:x+3≤12

∃!x∈R⁺ :x²=42

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∀x∈M: A(x).

∃x∈M: A(x).

The statement “There is only onexinMsuch thatA(x)holds.” is shortened to

∃!x∈M: A(x).

Example

∀x∈R:|x| ≥0

∃x∈Q:x+3≤12

∃!x∈R⁺ :x²=42

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∀x∈M: A(x).

∃x∈M: A(x).

The statement “There is only onexinM such thatA(x)holds.”

is shortened to

∃!x∈M: A(x).

Example

∀x∈R:|x| ≥0

∃x∈Q:x+3≤12

∃!x∈R⁺ :x²=42

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∀x∈M: A(x).

∃x∈M: A(x).

The statement “There is only onexinM such thatA(x)holds.”

is shortened to

∃!x∈M: A(x).

Example

∀x∈R:|x| ≥0

∃x∈Q:x+3≤12

∃!x∈R⁺ :x²=42

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IfA(x),x∈M andB(x),x∈Mare predicates, then

∀x∈M,B(x) :A(x) means ∀x∈M: (B(x)⇒A(x)),

∃x∈M,B(x) : A(x) means ∃x∈M: (A(x) & B(x)). Example

∀x∈R,x≥ −1:1+2x≤(1+x)²

∃x∈R,x≥0:x≥x²

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IfA(x),x∈M andB(x),x∈Mare predicates, then

∀x∈M,B(x) :A(x) means ∀x∈M: (B(x)⇒A(x)),

∃x∈M,B(x) : A(x) means ∃x∈M: (A(x) & B(x)).

Example

∀x∈R,x≥ −1:1+2x≤(1+x)²

∃x∈R,x≥0:x≥x²

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Negations of the statements with quantifiers:

¬(∀x∈M: A(x)) is the same as ∃x∈M:¬A(x),

¬(∃x∈M: A(x)) is the same as ∀x∈M:¬A(x). Example

Find negation

∀x∈R,x≥ −1:1+2x≤(1+x)²

∀x∈R,∀y∈R,x≥0,y≥0: x+y

2 ≥√

xy

∃x∈R,x≥0:x≥x²

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Negations of the statements with quantifiers:

¬(∀x∈M: A(x)) is the same as ∃x∈M:¬A(x),

¬(∃x∈M: A(x)) is the same as ∀x∈M:¬A(x).

Example Find negation

∀x∈R,x≥ −1:1+2x≤(1+x)²

∀x∈R,∀y∈R,x≥0,y≥0: x+y

2 ≥√

xy

∃x∈R,x≥0:x≥x²

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Methods of proofs

direct proof indirect proof

proof by contradiction mathematical induction

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Methods of proofs

direct proof indirect proof

proof by contradiction mathematical induction

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Induction

Exercise

n

X

i=1

(2i−1) =n²

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Rational numbers

The set of natural numbers

N={1,2,3,4, . . .}

The set of integers

Z=N∪ {0} ∪ {−n; n∈N}={. . . ,−2,−1,0,1,2, . . .}

The set of rational numbers Q=

p

q; p∈Z,q∈N

, where ^p_q¹

1 = ^p_q²

2 if and only ifp₁·q₂ =p₂·q₁.

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Rational numbers

N={1,2,3,4, . . .}

The set of integers

Z=N∪ {0} ∪ {−n; n∈N}={. . . ,−2,−1,0,1,2, . . .}

p

q; p∈Z,q∈N

, where ^p_q¹

1 = ^p_q²

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Rational numbers

N={1,2,3,4, . . .}

The set of integers

Z=N∪ {0} ∪ {−n; n∈N}={. . . ,−2,−1,0,1,2, . . .}

p

q; p∈Z,q∈N

, where ^p_q¹

1 = ^p_q²

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Rational numbers

N={1,2,3,4, . . .}

The set of integers

Z=N∪ {0} ∪ {−n; n∈N}={. . . ,−2,−1,0,1,2, . . .}

p

q; p∈Z,q∈N

, where ^p_q¹

1 = ^p_q²

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Rational numbers

N={1,2,3,4, . . .}

The set of integers

Z=N∪ {0} ∪ {−n; n∈N}={. . . ,−2,−1,0,1,2, . . .}

p

q; p∈Z,q∈N

, where ^p_q¹

1 = ^p_q²

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Rational numbers

N={1,2,3,4, . . .}

The set of integers

Z=N∪ {0} ∪ {−n; n∈N}={. . . ,−2,−1,0,1,2, . . .}

p

q; p∈Z,q∈N

, where ^p_q¹

1 = ^p_q²

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Rational numbers

N={1,2,3,4, . . .}

The set of integers

Z=N∪ {0} ∪ {−n; n∈N}={. . . ,−2,−1,0,1,2, . . .}

The set of rational numbers

Q= p

q; p∈Z,q∈N

, where ^p_q¹

1 = ^p_q²

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Rational numbers

N={1,2,3,4, . . .}

The set of integers

Z=N∪ {0} ∪ {−n; n∈N}={. . . ,−2,−1,0,1,2, . . .}

The set of rational numbers

Q= p

q; p∈Z,q∈N

, where ^p_q¹

1 = ^p_q²

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Rational numbers

N={1,2,3,4, . . .}

The set of integers

Z=N∪ {0} ∪ {−n; n∈N}={. . . ,−2,−1,0,1,2, . . .}

p

q; p∈Z,q∈N

, where ^p_q¹

1 = ^p_q²

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Real numbers

By the set of real numbersRwe will understand a set on which there are operations ofadditionandmultiplication(denoted by +and·), and a relation ofordering(denoted by≤), such that it has the following three groups of properties.

I. The properties of addition and multiplication and their relationships.

II. The relationships of the ordering and the operations of addition and multiplication.

III. The infimum axiom.

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Real numbers

By the set of real numbersRwe will understand a set on which there are operations ofadditionandmultiplication(denoted by +and·), and a relation ofordering(denoted by≤), such that it has the following three groups of properties.

I. The properties of addition and multiplication and their relationships.

II. The relationships of the ordering and the operations of addition and multiplication.

III. The infimum axiom.

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The properties of addition and multiplication and their relationships:

∀x,y∈R: x+y=y+x(commutativity of addition),

∀x,y,z∈R: x+ (y+z) = (x+y) +z(associativity), There is an element inR(denoted by 0 and called azero element), such thatx+0=xfor everyx∈R,

∀x∈R∃y∈R: x+y=0 (yis called thenegativeofx, suchyis only one, denoted by−x),

∀x,y∈R: x·y=y·x(commutativity),

∀x,y,z∈R: x·(y·z) = (x·y)·z(associativity),

There is a non-zero element inR(calledidentity, denoted by 1), such that 1·x=xfor everyx∈R,

∀x∈R\ {0} ∃y∈R:x·y=1 (suchyis only one, denoted byx⁻¹or ¹_x),

∀x,y,z∈R: (x+y)·z=x·z+y·z(distributivity).

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The properties of addition and multiplication and their relationships:

∀x,y∈R: x+y=y+x(commutativity of addition),

∀x,y,z∈R: x+ (y+z) = (x+y) +z(associativity), There is an element inR(denoted by 0 and called azero element), such thatx+0=xfor everyx∈R,

∀x∈R∃y∈R: x+y=0 (yis called thenegativeofx, suchyis only one, denoted by−x),

∀x,y∈R: x·y=y·x(commutativity),

∀x,y,z∈R: x·(y·z) = (x·y)·z(associativity),

There is a non-zero element inR(calledidentity, denoted by 1), such that 1·x=xfor everyx∈R,

∀x∈R\ {0} ∃y∈R:x·y=1 (suchyis only one, denoted byx⁻¹or ¹_x),

∀x,y,z∈R: (x+y)·z=x·z+y·z(distributivity).

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The relationships of the ordering and the operations of addition and multiplication:

∀x,y,z∈R: (x≤y & y≤z)⇒x≤z(transitivity),

∀x,y∈R: (x≤y & y≤x)⇒x=y(weak antisymmetry),

∀x,y∈R: x≤y∨y≤x,

∀x,y,z∈R: x≤y⇒x+z≤y+z,

∀x,y∈R: (0≤x & 0≤y)⇒0≤x·y.

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The relationships of the ordering and the operations of addition and multiplication:

∀x,y,z∈R: (x≤y & y≤z)⇒x≤z(transitivity),

∀x,y∈R: (x≤y & y≤x)⇒x=y(weak antisymmetry),

∀x,y∈R: x≤y∨y≤x,

∀x,y,z∈R: x≤y⇒x+z≤y+z,

∀x,y∈R: (0≤x & 0≤y)⇒0≤x·y.

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Definition

We say that the setM ⊂Risbounded from belowif there exists a numbera∈Rsuch that for eachx∈Mwe havex≥a.

Such a numberais called alower boundof the setM.

Analogously we define the notions of aset bounded from above and anupper bound. We say that a setM ⊂Risboundedif it is bounded from above and below.

Exercise

Which sets are bounded from below? Bounded from above? Bounded?

A N

B {1,¹₂,¹₃,¹₄,¹₅, . . .} C R\Q∩(−3,2]

D {x∈R:x< π}

E (−∞,−1)∪ {0} ∪[1,∞)

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Definition

Exercise

A N

B {1,¹₂,¹₃,¹₄,¹₅, . . .} C R\Q∩(−3,2]

D {x∈R:x< π}

E (−∞,−1)∪ {0} ∪[1,∞)

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Definition

Analogously we define the notions of aset bounded from above and anupper bound.

We say that a setM ⊂Risboundedif it is bounded from above and below.

Exercise

A N

B {1,¹₂,¹₃,¹₄,¹₅, . . .} C R\Q∩(−3,2]

D {x∈R:x< π}

E (−∞,−1)∪ {0} ∪[1,∞)

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Definition

Exercise

Which sets are bounded from below? Bounded from above?

Bounded?

A N

B {1,¹₂,¹₃,¹₄,¹₅, . . .}

C R\Q∩(−3,2]

D {x∈R:x< π}

E (−∞,−1)∪ {0} ∪[1,∞)

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The infimum axiom:

LetMbe a non-empty set bounded from below. Then there exists a unique numberg∈Rsuch that

(i) ∀x∈M: x≥g,

(ii) ∀g^′ ∈R,g^′ >g∃x∈M: x<g^′.

The numbergis denoted byinfMand is called theinfimumof the setM.

Figure:

https://mathspandorabox.wordpress.com/2016/03/11/the-difference- between-supremum-and-infimum-equivalent-and-equal-set/

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The infimum axiom:

(i) ∀x∈M: x≥g,

(ii) ∀g^′ ∈R,g^′ >g∃x∈M: x<g^′.

Figure:

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The infimum axiom:

(i) ∀x∈M: x≥g,

(ii) ∀g^′ ∈R,g^′ >g∃x∈M: x<g^′.

Figure:

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Remark

The infimum axiom says that every non-empty set bounded from below has infimum.

The infimum of the setMis its greatest lower bound. The real numbers exist and are uniquely determined by the properties I–III.

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Remark

The infimum of the setMis its greatest lower bound.

The real numbers exist and are uniquely determined by the properties I–III.

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Remark

The infimum of the setMis its greatest lower bound.

The real numbers exist and are uniquely determined by the properties I–III.

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The following hold:

(i) ∀x∈R: x·0=0·x=0, (ii) ∀x∈R: −x= (−1)·x,

(iii) ∀x,y∈R: xy=0⇒(x=0∨y=0), (iv) ∀x∈R∀n∈N: x⁻ⁿ= (x⁻¹)ⁿ,

(v) ∀x,y∈R: (x>0∧y>0)⇒xy>0,

(vi) ∀x∈R,x≥0∀y∈R,y≥0∀n∈N: x<y⇔xⁿ <yⁿ.

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Leta,b∈R,a≤b. We denote:

Anopen interval(a,b) ={x∈R; a<x<b}, Aclosed interval[a,b] ={x∈R; a≤x≤b}, Ahalf-open interval[a,b) ={x∈R; a≤x<b}, Ahalf-open interval(a,b] ={x∈R; a<x≤b}.

The pointais called theleft endpoint of the interval, The point bis called theright endpoint of the interval. A point in the interval which is not an endpoint is called aninner point of the interval.

Unbounded intervals:

(a,+∞) = {x∈R; a<x}, (−∞,a) ={x∈R; x<a}, analogically(−∞,a],[a,+∞)and(−∞,+∞).

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Label the Venn diagram withN,Q,Z,R,R\Q.

We haveN⊂Z⊂Q⊂R. If we transfer the addition and multiplication fromRto the above sets, we obtain the usual operations on these sets.

A real number that is not rational is calledirrational. The set R\Qis called theset of irrational numbers.

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Label the Venn diagram withN,Q,Z,R,R\Q.

We haveN⊂Z⊂Q⊂R. If we transfer the addition and multiplication fromRto the above sets, we obtain the usual operations on these sets.

A real number that is not rational is calledirrational. The set

\ is called theset of irrational numbers.

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Consequences of the infimum axiom

Definition

LetM ⊂R. A numberG∈Rsatisfying (i) ∀x∈M: x≤G,

(ii) ∀G^′ ∈R,G^′ <G∃x∈M: x>G^′, is called asupremumof the setM.

Theorem 1 (Supremum theorem)

Let M ⊂Rbe a non-empty set bounded from above. Then there exists a unique supremum of the set M.

The supremum of the setMis denoted bysupM. The following holds:supM =−inf(−M).

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Consequences of the infimum axiom

Definition

The supremum of the setMis denoted bysupM. The following holds:supM =−inf(−M).

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Consequences of the infimum axiom

Definition

The supremum of the setMis denoted bysupM.

The following holds:supM =−inf(−M).

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Consequences of the infimum axiom

Definition

The supremum of the setMis denoted bysupM.

The following holds:supM =−inf(−M).

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Definition

LetM ⊂R. We say thatais amaximumof the setM(denoted bymaxM) ifais an upper bound ofManda∈M.

Analogously we define aminimumofM, denoted byminM.

Exercise

Find infimum, minimum, maximum and supremum:

1. {1,2,3,4}

2. [−2,3]

3. (−2,3) 4. (−2,3]

5. [−2,−1)∪(0,25]

6. (−7,−0)∪(1,2) 7. [0,∞)

8. {1,¹₂,¹₃,¹₄, . . .}

9. N

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Theorem 2 (Archimedean property)

For every x∈Rthere exists n∈Nsatisfying n >x.

Theorem 3 (existence of an integer part)

For every r∈Rthere exists aninteger partof r, i.e. a number k∈Zsatisfying k ≤r<k+1. The integer part of r is

determined uniquely and it is denoted by[r].

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Theorem 2 (Archimedean property)

For every x∈Rthere exists n∈Nsatisfying n >x.

Theorem 3 (existence of an integer part)

For every r∈Rthere exists aninteger partof r, i.e. a number k∈Zsatisfying k ≤r<k+1. The integer part of r is

determined uniquely and it is denoted by[r].

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Theorem 4 (nth root)

For every x∈[0,+∞)and every n∈Nthere exists a unique y∈[0,+∞)satisfying yⁿ =x.

Theorem 5 (density ofQandR\Q)

Let a,b∈R, a<b. Then there exist r∈Qsatisfying a<r<b and s∈R\Qsatisfying a<s<b.

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Theorem 4 (nth root)

For every x∈[0,+∞)and every n∈Nthere exists a unique y∈[0,+∞)satisfying yⁿ =x.

Theorem 5 (density ofQandR\Q)

Let a,b∈R, a<b. Then there exist r∈Qsatisfying a<r<b and s∈R\Qsatisfying a<s<b.

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II. Limit of a sequence

Definition

Suppose that to each natural numbern∈Nwe assign a real numbera_n. Then we say that{an}^∞_n=1 is asequenceof real numbers. The numberanis called thenth memberof this sequence.

A sequence{an}^∞_n=1is equal to a sequence{bn}^∞_n=1ifa_n =b_n holds for everyn∈N.

By theset of all members of the sequence{an}^∞_n=1 we understand the set

{x∈R; ∃n∈N: an =x}.

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II. Limit of a sequence

Definition

Suppose that to each natural numbern∈Nwe assign a real numbera_n. Then we say that{an}^∞_n=1 is asequenceof real numbers.

The numberanis called thenth memberof this sequence.

{x∈R; ∃n∈N: an =x}.

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II. Limit of a sequence

Definition

{x∈R; ∃n∈N: an =x}.

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II. Limit of a sequence

Definition

{x∈R; ∃n∈N: an =x}.

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II. Limit of a sequence

Definition

{x∈R; ∃n∈N: an =x}.

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Posloupnost {1/n}

(105)

Posloupnost {(–1)^n}

(106)

Posloupnost {n}

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Posloupnost {P_n}

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Definition

We say that a sequence{an}is

bounded from aboveif the set of all members of this sequence is bounded from above,

bounded from belowif the set of all members of this sequence is bounded from below,

boundedif the set of all members of this sequence is bounded.

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Definition

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Definition

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Definition

We say that a sequence{a_n}is

increasingifa_n <a_n+1 for everyn∈N,

decreasingifa_n >a_n+1 for everyn∈N, non-decreasingifa_n ≤a_n+1for everyn∈N, non-increasingifa_n ≥a_n+1for everyn∈N.

A sequence{an}ismonotoneif it satisfies one of the conditions above. A sequence{an}isstrictly monotoneif it is increasing or decreasing.

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Definition

increasingifa_n <a_n+1 for everyn∈N, decreasingifa_n >a_n+1 for everyn∈N,

non-decreasingifa_n ≤a_n+1for everyn∈N, non-increasingifa_n ≥a_n+1for everyn∈N.

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Definition

increasingifa_n <a_n+1 for everyn∈N, decreasingifa_n >a_n+1 for everyn∈N, non-decreasingifa_n ≤a_n+1for everyn∈N,

non-increasingifa_n ≥a_n+1for everyn∈N.

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Definition

increasingifa_n <a_n+1 for everyn∈N, decreasingifa_n >a_n+1 for everyn∈N, non-decreasingifa_n ≤a_n+1for everyn∈N, non-increasingifa_n ≥a_n+1for everyn∈N.

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Definition

A sequence{an}ismonotoneif it satisfies one of the conditions above.

A sequence{an}isstrictly monotoneif it is increasing or decreasing.

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Definition

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Definition

Let{an}and{bn}be sequences of real numbers.

By thesum of sequences{an}and{bn}we understand a sequence{a_n+b_n}.

Analogously we define adifferenceand aproduct of sequences.

Suppose all the members of the sequence{bn}are

non-zero. Then by thequotient of sequences{an}and{bn} we understand a sequence{^a_bⁿ

n}.

Ifλ∈R, then by theλ-multiple of the sequence{an}we understand a sequence{λa_n}.

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Definition

Let{an}and{bn}be sequences of real numbers.

By thesum of sequences{an}and{bn}we understand a sequence{a_n+b_n}.

Analogously we define adifferenceand aproduct of sequences.

Suppose all the members of the sequence{bn}are

non-zero. Then by thequotient of sequences{an}and{bn} we understand a sequence{^a_bⁿ

n}.

Ifλ∈R, then by theλ-multiple of the sequence{an}we understand a sequence{λa_n}.