Math 316 Introduction to real analysis I Lecture 1 316 Introduction to real analysis I Lecture 1...

Math 316Introduction to real analysis I

Lecture 1

Christopher Davis

Christopher Davis Math 316Introduction to real analysis ILecture 1 1 / 16

Who am I?

Christopher Davis,Office: 512 Hibbard Humanities Hall

eMail: [email protected] hours: Monday, Wednesday,

Thursday, Friday 1-2 PM(also by appointment,

but please give me warning)


Syllabus:


Outline of the class

The Goal of Math 316 and 317 are to discuss the basic facts which makeCalculus work. Our goals will be:

Basic properties of the Real lineI R is a complete ordered field.

Sequences and their limits.

The Cauchy criterion for convergence.

Bounded sequences and the Bolzano-Weierstrass Theorem.

Infinite series, convergence, absolute convergence.

Functions and their limits

Continuity

Let’s break right now so that you can tell me how much you know aboutset theory.


Naive set theory

In the formal study of set theory, the word set is not defined.

Rather sets are assumed to follow some basic axiom called the Zermelo -Fraenkel axioms of set theory together with another axiom called theAxiom of choiceWe will not linger on this. For us a set is a collection of things, calledelementsIf x is an element of the set A, then we say that x ∈ A. Otherwise wewrite x /∈ Aexample The Natural numbers N := {1, 2, 3, 4, . . . } form a set. 457 ∈ N,0 /∈ N.Break to look at the preliminaries quiz.


Naive set theory

In the formal study of set theory, the word set is not defined.Rather sets are assumed to follow some basic axiom called the Zermelo -Fraenkel axioms of set theory together with another axiom called theAxiom of choice

We will not linger on this. For us a set is a collection of things, calledelementsIf x is an element of the set A, then we say that x ∈ A. Otherwise wewrite x /∈ Aexample The Natural numbers N := {1, 2, 3, 4, . . . } form a set. 457 ∈ N,0 /∈ N.Break to look at the preliminaries quiz.


Naive set theory

In the formal study of set theory, the word set is not defined.Rather sets are assumed to follow some basic axiom called the Zermelo -Fraenkel axioms of set theory together with another axiom called theAxiom of choiceWe will not linger on this. For us a set is a collection of things, calledelements

If x is an element of the set A, then we say that x ∈ A. Otherwise wewrite x /∈ Aexample The Natural numbers N := {1, 2, 3, 4, . . . } form a set. 457 ∈ N,0 /∈ N.Break to look at the preliminaries quiz.


Naive set theory

In the formal study of set theory, the word set is not defined.Rather sets are assumed to follow some basic axiom called the Zermelo -Fraenkel axioms of set theory together with another axiom called theAxiom of choiceWe will not linger on this. For us a set is a collection of things, calledelementsIf x is an element of the set A, then we say that x ∈ A. Otherwise wewrite x /∈ Aexample The Natural numbers N := {1, 2, 3, 4, . . . } form a set. 457 ∈ N,0 /∈ N.

Break to look at the preliminaries quiz.


Naive set theory

In the formal study of set theory, the word set is not defined.Rather sets are assumed to follow some basic axiom called the Zermelo -Fraenkel axioms of set theory together with another axiom called theAxiom of choiceWe will not linger on this. For us a set is a collection of things, calledelementsIf x is an element of the set A, then we say that x ∈ A. Otherwise wewrite x /∈ Aexample The Natural numbers N := {1, 2, 3, 4, . . . } form a set. 457 ∈ N,0 /∈ N.Break to look at the preliminaries quiz.


Relations between sets

Sets can can contain each other, not as elements but as subsets.

We say that a set A is contained in another set B if every element of A isalso an element of B. We abbreviate this as A ⊆ B.


Two sets are called equal if they contain the same elements.One good way of showing two sets A and B are equal is by showing thatA ⊆ B and B ⊆ A.Examples:Let’s show that {−1, 1} is equal to {x : x2 = 1}.The notation {x : x2 = 1} should be read as “The set of all x satisfyingthat x2 = 1.”



Sets can can contain each other, not as elements but as subsets.We say that a set A is contained in another set B if every element of A isalso an element of B. We abbreviate this as A ⊆ B.







Two sets are called equal if they contain the same elements.

One good way of showing two sets A and B are equal is by showing thatA ⊆ B and B ⊆ A.Examples:Let’s show that {−1, 1} is equal to {x : x2 = 1}.The notation {x : x2 = 1} should be read as “The set of all x satisfyingthat x2 = 1.”





Two sets are called equal if they contain the same elements.One good way of showing two sets A and B are equal is by showing thatA ⊆ B and B ⊆ A.

Examples:Let’s show that {−1, 1} is equal to {x : x2 = 1}.The notation {x : x2 = 1} should be read as “The set of all x satisfyingthat x2 = 1.”


The notion on the previous slide: how do you describesets?

One good way of building sets is by making up some conditions thatdetermine weather an element is in that set.

we will use the notation {x : conditions} to describe the set of all x whichsatisfy some conditions.For example {x : x ∈ R, 0 < x , and x ≤ 3} is the set of all Real numberswhich are greater than 0 and less than or equal to 3.Who recognizes this set?We are sometimes no so rigid in following this notation. examples2 ·N := {2 · n : n ∈ N} “The set of all numbers of the form 2 · n where n isan integer.Q := {p/q : p, q ∈ Z and q 6= 0} “The set of all numbers of the form 2 · nwhere n is an integer.



One good way of building sets is by making up some conditions thatdetermine weather an element is in that set.we will use the notation {x : conditions} to describe the set of all x whichsatisfy some conditions.

For example {x : x ∈ R, 0 < x , and x ≤ 3} is the set of all Real numberswhich are greater than 0 and less than or equal to 3.Who recognizes this set?We are sometimes no so rigid in following this notation. examples2 ·N := {2 · n : n ∈ N} “The set of all numbers of the form 2 · n where n isan integer.Q := {p/q : p, q ∈ Z and q 6= 0} “The set of all numbers of the form 2 · nwhere n is an integer.



One good way of building sets is by making up some conditions thatdetermine weather an element is in that set.we will use the notation {x : conditions} to describe the set of all x whichsatisfy some conditions.For example {x : x ∈ R, 0 < x , and x ≤ 3} is the set of all Real numberswhich are greater than 0 and less than or equal to 3.

Who recognizes this set?We are sometimes no so rigid in following this notation. examples2 ·N := {2 · n : n ∈ N} “The set of all numbers of the form 2 · n where n isan integer.Q := {p/q : p, q ∈ Z and q 6= 0} “The set of all numbers of the form 2 · nwhere n is an integer.



One good way of building sets is by making up some conditions thatdetermine weather an element is in that set.we will use the notation {x : conditions} to describe the set of all x whichsatisfy some conditions.For example {x : x ∈ R, 0 < x , and x ≤ 3} is the set of all Real numberswhich are greater than 0 and less than or equal to 3.Who recognizes this set?

We are sometimes no so rigid in following this notation. examples2 ·N := {2 · n : n ∈ N} “The set of all numbers of the form 2 · n where n isan integer.Q := {p/q : p, q ∈ Z and q 6= 0} “The set of all numbers of the form 2 · nwhere n is an integer.



One good way of building sets is by making up some conditions thatdetermine weather an element is in that set.we will use the notation {x : conditions} to describe the set of all x whichsatisfy some conditions.For example {x : x ∈ R, 0 < x , and x ≤ 3} is the set of all Real numberswhich are greater than 0 and less than or equal to 3.Who recognizes this set?We are sometimes no so rigid in following this notation. examples

2 ·N := {2 · n : n ∈ N} “The set of all numbers of the form 2 · n where n isan integer.Q := {p/q : p, q ∈ Z and q 6= 0} “The set of all numbers of the form 2 · nwhere n is an integer.



One good way of building sets is by making up some conditions thatdetermine weather an element is in that set.we will use the notation {x : conditions} to describe the set of all x whichsatisfy some conditions.For example {x : x ∈ R, 0 < x , and x ≤ 3} is the set of all Real numberswhich are greater than 0 and less than or equal to 3.Who recognizes this set?We are sometimes no so rigid in following this notation. examples2 ·N := {2 · n : n ∈ N} “The set of all numbers of the form 2 · n where n isan integer.

Q := {p/q : p, q ∈ Z and q 6= 0} “The set of all numbers of the form 2 · nwhere n is an integer.



One good way of building sets is by making up some conditions thatdetermine weather an element is in that set.we will use the notation {x : conditions} to describe the set of all x whichsatisfy some conditions.For example {x : x ∈ R, 0 < x , and x ≤ 3} is the set of all Real numberswhich are greater than 0 and less than or equal to 3.Who recognizes this set?We are sometimes no so rigid in following this notation. examples2 ·N := {2 · n : n ∈ N} “The set of all numbers of the form 2 · n where n isan integer.Q := {p/q : p, q ∈ Z and q 6= 0} “The set of all numbers of the form 2 · nwhere n is an integer.


Set operations: New sets from oldSuppose that you have sets A and B.The intersection of A and B is defined by

A ∩ B = {x : x ∈ A and x ∈ B}

The union of A and B is defined by

A ∪ B = {x : x ∈ A or x ∈ B}

A remark: When a mathematician uses the word “or” she / he meansthat one of the two statements is true OR THAT THEY ARE BOTHTRUE!The set with no elements, The empty set is denoted ∅.If A ∩ B = ∅ then A and B are called disjoint.Exercises:

Complete some quiz problems

For all sets A and B, prove that A ∩ B ⊆ A ⊆ A ∪ B.



A ∩ B = {x : x ∈ A and x ∈ B}


A ∪ B = {x : x ∈ A or x ∈ B}

A remark: When a mathematician uses the word “or” she / he meansthat one of the two statements is true OR THAT THEY ARE BOTHTRUE!

The set with no elements, The empty set is denoted ∅.If A ∩ B = ∅ then A and B are called disjoint.Exercises:


For all sets A and B, prove that A ∩ B ⊆ A ⊆ A ∪ B.



A ∩ B = {x : x ∈ A and x ∈ B}


A ∪ B = {x : x ∈ A or x ∈ B}

A remark: When a mathematician uses the word “or” she / he meansthat one of the two statements is true OR THAT THEY ARE BOTHTRUE!The set with no elements, The empty set is denoted ∅.If A ∩ B = ∅ then A and B are called disjoint.Exercises:


For all sets A and B, prove that A ∩ B ⊆ A ⊆ A ∪ B.Christopher Davis Math 316Introduction to real analysis ILecture 1 8 / 16

Facts about intersection and union

Theorem

Let A, B and C be sets. then

(Idempotency) A ∪ A = A ∩ A = A

(commutatity) A ∪ B = B ∪ A, A ∩ B = B ∩ A

(associativity) A ∩ (B ∩ C ) = (A ∩ B) ∩ C,A ∪ (B ∪ C ) = (A ∪ B) ∪ C,

(distributativity) A ∩ (B ∪ C ) = (A ∩ B) ∪ (A ∩ C ),A ∪ (B ∩ C ) = (A ∪ B) ∩ (A ∪ C ),

I’ll prove that A = A ∩ A


complementary sets and De Morgan’s Law

The complement of a set B in a set A is written as

A− B := {x : x ∈ A and x /∈ B}.

The notation comes from thinking about taking all of the elements of Aand removing those in BLet’s complete another quiz problem.

Proposition (De Morgan’s Law)

For sets A, B, and C,

A− (B ∪ C ) = (A− B) ∩ (A− C )

A− (B ∩ C ) = (A− B) ∪ (A− C )

I’ll give the proof of one and leave the other to group work.


The Cartesian Product

For two sets A and B, the cartesian product A× B is the set of all orderedpairs (a, b) such that a ∈ A and b ∈ B.

A× B = {(a, b) : a ∈ A and b ∈ B}

Look back at the group work

For example, for the function f (x) = x2 + 1, the graph of f ,{(x , x2 + 1) : x ∈ R} is a subset of R× R. This is a good reason to careabout the Cartesian product. Its where graphs of functions live.


Functions.For sets A and B the string f : A→ B should be read f is a function formA to B.What does this mean? What is a function?

Intuitive definition: A function f : A→ B is a rule sending eachelements of A to a single element of B.This is no better as a definition. What is a “rule”? But we are workingintuitively and naively today. Friday we’ll start being really rigorous.example f : {1, 2, 3, 4} → {♥, ?, †} defined by 1 7→ ♥, 2 7→ ♥, 3 7→ ?,4 7→ ♥ is a function .

Each element of {1, 2, 3, 4} is sent to a single element of {♥, ?, †}.Not every element of {♥, ?, †} is hit, but that’s OK.

some elements of {♥, ?, †} are hit twice, but that’s OK, too.

Nonexamples: Why are these not functions?

f : {1, 2, 3, 4} → {♥, ?, †} 1 7→ ♥, 3 7→ ?, 4 7→ ♥.

f : {1, 2, 3, 4} → {♥, ?, †} 1 7→ ♥, 2 7→ ♥, 2 7→ ?, 3 7→ ?, 4 7→ ♥.

f : {1, 2, 3, 4} → {♥, ?, †} 1 7→ ♥, 2 7→ ♥, 3 7→ ?, 4 7→ ♠.


Functions.For sets A and B the string f : A→ B should be read f is a function formA to B.What does this mean? What is a function?Intuitive definition: A function f : A→ B is a rule sending eachelements of A to a single element of B.

This is no better as a definition. What is a “rule”? But we are workingintuitively and naively today. Friday we’ll start being really rigorous.example f : {1, 2, 3, 4} → {♥, ?, †} defined by 1 7→ ♥, 2 7→ ♥, 3 7→ ?,4 7→ ♥ is a function .




f : {1, 2, 3, 4} → {♥, ?, †} 1 7→ ♥, 3 7→ ?, 4 7→ ♥.

f : {1, 2, 3, 4} → {♥, ?, †} 1 7→ ♥, 2 7→ ♥, 2 7→ ?, 3 7→ ?, 4 7→ ♥.

f : {1, 2, 3, 4} → {♥, ?, †} 1 7→ ♥, 2 7→ ♥, 3 7→ ?, 4 7→ ♠.


Functions.For sets A and B the string f : A→ B should be read f is a function formA to B.What does this mean? What is a function?Intuitive definition: A function f : A→ B is a rule sending eachelements of A to a single element of B.This is no better as a definition. What is a “rule”? But we are workingintuitively and naively today. Friday we’ll start being really rigorous.

example f : {1, 2, 3, 4} → {♥, ?, †} defined by 1 7→ ♥, 2 7→ ♥, 3 7→ ?,4 7→ ♥ is a function .




f : {1, 2, 3, 4} → {♥, ?, †} 1 7→ ♥, 3 7→ ?, 4 7→ ♥.

f : {1, 2, 3, 4} → {♥, ?, †} 1 7→ ♥, 2 7→ ♥, 2 7→ ?, 3 7→ ?, 4 7→ ♥.

f : {1, 2, 3, 4} → {♥, ?, †} 1 7→ ♥, 2 7→ ♥, 3 7→ ?, 4 7→ ♠.


Functions.For sets A and B the string f : A→ B should be read f is a function formA to B.What does this mean? What is a function?Intuitive definition: A function f : A→ B is a rule sending eachelements of A to a single element of B.This is no better as a definition. What is a “rule”? But we are workingintuitively and naively today. Friday we’ll start being really rigorous.example f : {1, 2, 3, 4} → {♥, ?, †} defined by 1 7→ ♥, 2 7→ ♥, 3 7→ ?,4 7→ ♥ is a function .




f : {1, 2, 3, 4} → {♥, ?, †} 1 7→ ♥, 3 7→ ?, 4 7→ ♥.

f : {1, 2, 3, 4} → {♥, ?, †} 1 7→ ♥, 2 7→ ♥, 2 7→ ?, 3 7→ ?, 4 7→ ♥.

f : {1, 2, 3, 4} → {♥, ?, †} 1 7→ ♥, 2 7→ ♥, 3 7→ ?, 4 7→ ♠.


domain, range, image and pre-image

Let f : A→ B be a function

A is called the domain of f . it is denoted with dom(f ) = D(f ) = A.The range of f is the set of all elements of B ’hit’ by frng(f ) = R(f ) = {b : for some a ∈ D(f ), f (a) = b}For a set X ⊆ D(f ), the image of X under f is given by

f [X ] := {b : for some x ∈ X , f (x) = b}

For a set Y ⊆ B, the pre-image of Y under f is given by

f −1[X ] := {a : f (a) ∈ Y }

Back to Quiz Problems!



Let f : A→ B be a functionA is called the domain of f . it is denoted with dom(f ) = D(f ) = A.

The range of f is the set of all elements of B ’hit’ by frng(f ) = R(f ) = {b : for some a ∈ D(f ), f (a) = b}For a set X ⊆ D(f ), the image of X under f is given by



f −1[X ] := {a : f (a) ∈ Y }




Let f : A→ B be a functionA is called the domain of f . it is denoted with dom(f ) = D(f ) = A.The range of f is the set of all elements of B ’hit’ by f

rng(f ) = R(f ) = {b : for some a ∈ D(f ), f (a) = b}For a set X ⊆ D(f ), the image of X under f is given by



f −1[X ] := {a : f (a) ∈ Y }




Let f : A→ B be a functionA is called the domain of f . it is denoted with dom(f ) = D(f ) = A.The range of f is the set of all elements of B ’hit’ by frng(f ) = R(f ) = {b : for some a ∈ D(f ), f (a) = b}

For a set X ⊆ D(f ), the image of X under f is given by



f −1[X ] := {a : f (a) ∈ Y }




Let f : A→ B be a functionA is called the domain of f . it is denoted with dom(f ) = D(f ) = A.The range of f is the set of all elements of B ’hit’ by frng(f ) = R(f ) = {b : for some a ∈ D(f ), f (a) = b}For a set X ⊆ D(f ), the image of X under f is given by



f −1[X ] := {a : f (a) ∈ Y }



injections, surjections, and bijections

Definition

A function f : A→ B is

injective or one-to-one if whenever x1 6= x2, f (x1) 6= f (x2).

surjective or onto if f [A] = B.

bijective or invertible if it is both invective and surjective.

Now let’s finish the quiz!

Proposition

if f : A→ B is a bijective function, then there is another functiong : B → A given by g(f (a)) = a. This new function is denoted f −1 and iscalled the inverse function.

Proof: See any Precalculus book.Why must f be bijective?


composition

For functions f : A→ B and g : B → C and x ∈ A the composition g ◦ fis defined by

(g ◦ f )(x) = g(f (x))

Proposition

if f : A→ B is bijective and g = f −1 : B → A is its inverse, then g ◦ f isthe identity map on A and f ◦ g is the identity map on B.

The identity map Id : A→ A on a set A is the function given by Id(x) = x


Group work:

For sets A, B, C , a (not necessarily invertible) function f : A→ B, andsubsets X ⊆ A and Y ⊆ B

Prove that A ∩ B ⊆ A ⊆ A ∪ B

Prove that A ∩ B = A− (A− B).

Prove that A ∩ (B ∪ C ) = (A ∪ B) ∩ (A ∪ C )

A− (B ∩ C ) = (A− B) ∪ (A− C )

If f is a function from A to B, X ⊆ A and Y ⊆ B, prove thatf −1[f [X ]] = X , and f [f −1[Y ]] ⊆ Y

When is f [f −1[Y ]] = Y ? When is the containment proper?


Math 316 Introduction to real analysis I Lecture 1 316 Introduction to real analysis I Lecture 1...

Documents

Transcript of Math 316 Introduction to real analysis I Lecture 1 316 Introduction to real analysis I Lecture 1...