Real Analysis (1)

Real Analysis (1)

Preliminaries

Sets

If an element x is in a set A, we write:

xA

and say that x is a member of A, or that x belongs to A. If x is not in A, we write:

xA

If every element of a set A also belongs to a set B, we say that A is a subset of B and write:

AB

We say that A is a proper subset of B if AB, but there is at least one element of B that is not in A. In this case we write:

AB

Definition 1.11: Equal Sets

Two sets A and B are said to be equal, we write A=B if they contain the same elements. Thus, to prove that sets A and B are equal, we must show that:

AB,BA

A set is normally defined by either listing its elements explicitly, or by specifying a property that determines the elements of the set. If P denotes a property that is meaningful and unambiguous for elements of a set S, then we write:

{xS;P(x)}

for the set of all elements x in S for which the property P is true.

De Morgan's Law

If A,B,C are sets, then:

A/(BC)=(A/B)(A/C) A/(BC)=(A/B)(A/C)

Where A/B denotes the complement of B relative to A:

A/B:={x:xAxB}

Unions and Interactions

For a finite collection of sets {A1,....,An}, their union is the set A consisting of all elements that belong to at least one of the sets Ak, and their intersection consists of all elements that belong to all of the sets {Ak}.

For an infinite collection of sets {A1,....,An,...}, their union is the set of elements that belong to at least one of the sets Ak:

n=1An={x:nN,s.t xAn}

Similarly, their intersection is the set of elements that belong to all of these sets An:

n=1An={x:nN,s.t xAn}

Functions

Definition 1.15: Cartesian Product

If A and B are nonempty sets, then the Cartesian product A×B of A and B is the set of all ordered pairs (a,b) with aA and bB. That is,

A×B:={(a,b):aA,bB}

General Definition

A function f from a set A into a set B is a rule of correspondence that assigns to each element xA an uniquely determined element f(x) in B.

Definition 1.1.6: Function as sets

Let A and B be sets. Then a function from A to B is a set f of ordered pairs in A×B such that for each aA there exists a unique bB with (a,b)f

The set A of first elements of a function f is called the domain of f and is often denoted by D(f). The set of all second elements in f is called the range of f and denoted as R(f). Notice that:

D(f)=A

but:

R(f)B

The notation:

f:AB

is often used to indicate that f is a function from A into B. We will also say that f is a mapping of A into B. If b=f(a), we often refer to b as the value of f at a, or as the image of a under f.

Direct and Inverse Images

Let f:AB be a function with domain D(f)=A and range R(f)B

Definition 1.1.7: Direct Image and Inverse Image

If E is a subset of A, then the direct image of E under f is the subset f(E) of B given by:

f(E):={f(x):xE}

If H is a subset of B, then the inverse image of H under f is the subset f1(H) of A given by:

f1(H):={xA:f(x)H}

Definition 1.1.9: Injection, Surjection and Bijection

Let f:AB be a function from A to B

  1. The function f is said to be Injective(one to one) if whenever x1x2, then f(x1)f(x2).
  2. The function f is said to be Surjective(to map A onto B) if whenever f(A)=B, that is if R(f)=B.
  3. The function f is said to be Bijective if f is both injective and surjective.

Definition 1.1.11: Inverse Functions

If f:AB is bijection of A onto B then:

g:={(b,a)B×A:(a,b)f}

is a function on B into A. This function is called inverse function of f and denoted as f1

Remark, f1 (inverse image) make sense even if f has no inverse function. However, if the inverse function f1 does exist, then f1(H) is the direct image of the set HB under f1

Definition 1.1.12: Composition of Functions

If f:AB and g:BC, and if R(f)D(g)=B, then the composition function gf is the function from AC:

(gf)(x):=g(f(x))

Theorem 1.1.14

Let f:AB and g:BC be functions and let H be a subset of C, then we have:

(gf)1(H)=f1(g1(H))

Restrictions of Functions

If f:AB is a function and if A1A, we can define a function f1:A1B by:

f1(x):=f(x)xA1

The function f1 is called the restriction of f to A1 denoted by f1=f|A1.

Finite and Infinite Sets

From a mathematical perspective, what we are doing is defining a bijective mapping between the set and a portion of the set of natural numbers. If the set is such that the counting does not terminate, such as the set of natural numbers itself, then we describe the set as being infinite.

Definition 1.3.1: Set Elements, finite, infinite

  1. The empty set is said to have 0 elements.
  2. If nN, a set S is said to have n elements if there exists a bijection from the set N:{1,2,....,n} onto S.
  3. A set S is said to be finite, if it is either empty or it has n elements for some nN.
  4. A set S is said to be infinite if it is not finite.
  5. A set T1 is finite if and only if there is a bijection from T1 onto another set T2 that is finite.

Uniqueness Theorem:

If S is a finite set, then the number of elements in S is a unique number in N.

Theorem 1.3.3

The set N of natural numbers is an infinite set.

Theorem 1.3.4

  1. If A is a set with m elements and B is a set with n elements and if AB=, then AB has m+n elements.
  2. If A is a set with mN elements and CA is a set with 1 element, then A/C is a set with m1 elements.
  3. If C is an infinite set and B is a finite set, then C/B is an infinite set.

Theorem 1.3.5

Suppose that S and T are sets and that TS:

  1. If S is a finite set, then T is a finite set.
  2. If T is an infinite set, then S is an infinite set.

Definition 1.3.6: Denumberable, Countable

  1. A set S is said to be denumberable (countably infinite) if there exists a bijection of N onto S.
  2. A set S is said to be countable if it is either finite or denumberable.
  3. A set S is said to be uncountable if it is not countable.

Theorem 1.3.8

The set N×N is denumberable

Theorem 1.3.9

Suppose that S and T are sets and that TS:

  1. If S is a countable set, then T is a countable set.
  2. If T is an uncountableset, then S is an uncountable sets.

Theorem 1.3.10

The following statements are equivalent:

  1. S is a countable set.
  2. There exists a surjection of N onto S.
  3. There exists an injection of S into N.

Theorem

The set Q of all rational numbers is denumberable.

Theorem 1.3.12

If Am is a countable set for each mN, then the union A:=m=1Am is countable.

Real Numbers

Least Upper Bounds (Supremum) and Greatest Lower Bounds (infimum)

Definition 1.1.3.1

A set AR is bounded above if there exists a number bR such that ab for all aA. The number b is called an upper bound for A. Similarly, the set A is bounded below if there exists a lower bound lR satisfying la for every aA.

Definition 1.1.3.2

A real number s is the least upper bound for a set AR if:

  1. s is an upper bound for A.
  2. If b is any upper bound for A, bs.

The least upper bound is also called supremum of the set A denoted as:

supA

The greatest lower bound is called infimum for A:

infA

infimum and supremum are unique for a set A and may or may not be elements in the set A.

Definition 1.1.3.4

A real number a0 is a maximum of the set A if a0 is an element of A and a0a for all aA. Similarly, a number a1 is a minimum of A if a1A and a1a for every aA.

Axiom of Completeness

Every nonempty set of real numbers that is bounded above has a least upper bound.

Lemma 1.1.3.8

Assume sR is an upper bound for a set AR. Then s=supA if and only if for every choice of ϵ>0, there exists an element aA satisfying s<a+ϵ.

Theorem 1.1.4.1: Nested Interval Property

For each nN, assume we are given a closed interval In=[an,bn]={xR:anxb}. Assume also that each In contains In+1. Then the resulting nested sequence of closed intervals:

I1I2I3....

has a nonempty intersection. That is:

n=1In0

Theorem 1.1.4.2: Archimedean Property

  1. Given any number xR, there exists an nN satisfying n>x.
  2. Given any real number y>0, there exists an nN satisfying 1n<y

Theorem 1.1.4.3: Density of Q in R

For every two real numbers a and b with a<b, there exists a rational number r satisfying a<r<b. In other words, rational numbers are dense in R.

Theorem 1.1.4.5

There exists a real number αR satisfying α2=x,xR0

Cantor's Diagonalizing Method

Theorem 1.1.6.1

The open interval (0,1)={xR:0<x<1} is uncountable

Power Sets

Given a set A, the power set P(A) refers to the collection of all subsets of A. It is important to understand that P(A) is itself considered a set whose elements are the different possible subsets of A.

Cantor's Theorem

Given any set A, there does not exist a function f:AP(A) that is onto. That is, no function satisfies R(f)=B.