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Chapter 1: The Real PWhite

Discussion

Some Preliminaries Chapter 1: The Real Numbers The Axiom of Completeness

Consequences of Completeness Peter W. White Cantor’s Theorem [email protected] Epilogue Initial development by Keith E. Emmert

Department of Tarleton State University

Fall 2011 / Real Anaylsis I Chapter 1: The Real Numbers Overview PWhite

Discussion

Some √ Preliminaries Discussion: The Irrationality of 2 The Axiom of Completeness Consequences of Some Preliminaries Completeness

Cantor’s Theorem

Epilogue The Axiom of Completeness

Consequences of Completeness

Cantor’s Theorem

Epilogue Chapter 1: The Real Numbers The Big Pythagorean Oopsie PWhite I The Pythagoreans worshiped and fractions Discussion of integers (rational numbers). Some Preliminaries I They believed that everything could be connected to The Axiom of Completeness an or rational . Consequences of I Around 500 BC,√ the Pythagoreans discovered the Completeness irrationality of 2. Cantor’s Theorem I The wheels came off the bus. Epilogue I began and is often driven by the study of things that caused the “wheels to come off the bus.” Chapter 1: The Real Numbers Basic Definitions PWhite

Discussion Definition 1

Some I The natural numbers are defined by Preliminaries

The Axiom of N = {1, 2, 3,...}. Completeness I The integers are defined by Consequences of Completeness Z = {..., −3, −2, −1, 0, 1, 2, 3,...}. Cantor’s Theorem I The rational numbers are defined by Epilogue p  = | p, q ∈ , q 6= 0 . Q q Z

Theorem 2 There is no whose square is 2. Proof: Chapter 1: The Real Numbers Questions and Observations PWhite √ √ I Thus 2 6∈ . Where does 2 “belong?” Discussion Q I We define irrational numbers as those that are not Some Preliminaries rational. √ The Axiom of I Fact: n p with p prime is irrational. Completeness √ n n Consequences of I Note that p is a root of f (x) = x − p. Completeness I Are there other irrational numbers that are not Cantor’s Theorem algebraic roots of polynomials with rational Epilogue coefficients? I The real numbers could be defined as Q with it’s “holes” filled in. I More about these things later. Chapter 1: The Real Numbers Overview PWhite

Discussion

Some √ Preliminaries Discussion: The Irrationality of 2 The Axiom of Completeness Consequences of Some Preliminaries Completeness

Cantor’s Theorem

Epilogue The Axiom of Completeness

Consequences of Completeness

Cantor’s Theorem

Epilogue Chapter 1: The Real Numbers Sets PWhite

Discussion Definition 3

Some I A set is a collection of any objects, called elements. Preliminaries The Axiom of I If an element x is a member of a set A we write Completeness x ∈ A. If x is not an element of A we write x 6∈ A. Consequences of Completeness I The , ∅, is the set that contains no Cantor’s Theorem elements. Epilogue I The union of two sets A and B is defined by

A ∪ B = {x | x ∈ A or x ∈ B (or both)}.

I The intersection of two sets A and B is defined by

A ∩ B = {x | x ∈ A and x ∈ B}. Chapter 1: The Real Numbers Sets PWhite

Discussion Definition 4

Some I We say A is a of B, A ⊆ B, if Preliminaries

The Axiom of Completeness x ∈ A =⇒ x ∈ B. Consequences of Completeness I Two sets A and B are equal, A = B, if A ⊆ B and Cantor’s Theorem B ⊆ A. Epilogue I A is a proper subset of B if A ⊆ B and there exists y ∈ B such that y 6∈ A. c I The complement of a set A ⊆ R, A , is defined by

c A = {x ∈ R | x 6∈ A}.

Remark 5 The definition of complement is dependent upon your choice of “universe” (for us, R). Chapter 1: The Real Numbers Tiny Examples PWhite

Discussion Example 6

Some Let E represent the even numbers and O the odd Preliminaries numbers. Let S = {r ∈ Q | |r 2| < 2}. The Axiom of Completeness I Z = E ∪ O. Consequences of Completeness I E ∩ O = ∅. Cantor’s Theorem I E = ∩ E. Epilogue Z I Z ∩ S = {−1, 0, 1}. I Let A be a set. ∅ ⊆ A. c I What is ∅ ? c I What is R ? c I What is Q ? c I What is Z ? Chapter 1: The Real Numbers Tiny Examples PWhite

Discussion Definition 7

Some Let A1, A2, A3,..., An,... be an infinite collection of sets. Preliminaries I A countable union of sets is The Axiom of Completeness ∞ Consequences of [ [ Completeness An = An = A1 ∪ A2 ∪ · · · ∪ An ∪ · · · . Cantor’s Theorem n=1 n∈N Epilogue I A countable intersection of sets is ∞ \ \ An = An = A1 ∩ A2 ∩ · · · ∩ An ∩ · · · . n=1 n∈N

I A (descending) nested chain of sets satisfies the condition

A1 ⊇ A2 ⊇ · · · ⊇ An ⊇ · · · . Chapter 1: The Real Numbers Example PWhite

Discussion Example 8 Some For each n ∈ N, define An = {n, n + 1, n + 2,...}. Verify Preliminaries the following statements. The Axiom of Completeness I A1 ⊇ A2 ⊇ · · · ⊇ An ⊇ · · · . Consequences of \ Completeness I An = ∅. Cantor’s Theorem n∈N Epilogue [ I An = A1. n∈N Chapter 1: The Real Numbers De Morgan’s Laws PWhite

Discussion Theorem 9

Some Let A and B be any sets. Then the following statements Preliminaries are true: The Axiom of Completeness 1. (A ∪ B)c = Ac ∩ Bc. Consequences of c c c Completeness 2. (A ∩ B) = A ∪ B . Cantor’s Theorem Epilogue The proof is an exercise. Chapter 1: The Real Numbers Functions PWhite

Discussion Definition 10

Some Let A and B be two sets. A function, f , from A to B is a Preliminaries rule or mapping that takes each element x ∈ A and The Axiom of Completeness associates with it a single element of B. Our notation is

Consequences of Completeness f : A 7→ B. Cantor’s Theorem Epilogue For x ∈ A, the expression f (x) is used to represent the element of B associated with x by f .

The set A is called the domain of f and the set B is called the codomain. The range or image of f is the set {y ∈ B | y = f (x) for some x ∈ A}.

Remark 11 If f : A 7→ B, then the range of f is a subset of the codomain, B, and need not equal B. Chapter 1: The Real Numbers Function Examples PWhite

Discussion Example 12 Some 2 Preliminaries I f : R 7→ R be defined by f (x) = x . Then the range of The Axiom of f is [0, ∞) (R. Completeness I Dirichlet’s Function: g : R 7→ R and is defined by Consequences of ( Completeness 1, x ∈ Q Cantor’s Theorem g(x) = The range if g is the set {0, 1}. 0, x 6∈ . Epilogue Q

Remark 13 When looking for counterexamples of statements about functions, try to remember Dirichlet’s Function. Chapter 1: The Real Numbers Preimages of Functions PWhite

Discussion Definition 14

Some Let f be a function satisfying f : D 7→ R and choose − Preliminaries B ⊆ R. Define the preimage of f , f 1(B) to be the set The Axiom of Completeness { ∈ | ( ) ∈ }. Consequences of x D f x B Completeness

Cantor’s Theorem Epilogue Example 15 Let the function f :[0, 4] 7→ R be given by f (x) = x2. Find the following preimages. −1 I f ([0, 1]) −1 I f ([−1, 1]) −1 I f (R) Chapter 1: The Real Numbers Some Preimage Theory PWhite

Discussion Theorem 16

Some Let D ⊆ R, suppose f : D 7→ R is a function and A, B ⊆ R Preliminaries be two sets. Prove the following statements The Axiom of −1 −1 −1 Completeness I f (A ∩ B) = f (A) ∩ f (B). Consequences of −1 −1 −1 Completeness I f (A ∪ B) = f (A) ∪ f (B). Cantor’s Theorem Proof: (Pick One) Epilogue Chapter 1: The Real Numbers Absolute Value Function PWhite

Discussion Definition 17

Some The absolute value of a number x ∈ R is given by the Preliminaries function ( The Axiom of x, x ≥ 0 Completeness |x| = Consequences of −x, x < 0. Completeness

Cantor’s Theorem Epilogue Remark 18 For all a, b, c ∈ R, 1. |ab| = |a||b| 2. |a + b| ≤ |a| + |b|.(The Triangle Inequality) 3. Common use of the Triangle Inequality:

|a − b| = |(a − c) + (c − b)| ≤ |a − c| + |c − b|. Chapter 1: The Real Numbers Logic and Proofs PWhite Some Types of Proofs: Discussion I Direct Proof – Begin with a true statement (often the Some Preliminaries hypothesis of your theorem) and use logical The Axiom of Completeness deductions to arrive at the conclusion of the

Consequences of statement. Completeness I Proof by Contradiction – Assume the hypothesis and Cantor’s Theorem the negation of the conclusion. The goal is to derive Epilogue a contradiction of some known statement. I Contrapositive Proof – Assume the negation of the conclusion and use logical deductions to arrive at the negation of the hypothesis. Chapter 1: The Real Numbers Logic and Proofs PWhite So, if you’re trying to prove A =⇒ B, then your typical Discussion options are: Some Preliminaries I Direct Proof: A =⇒ B. (Start with A and derive B). The Axiom of Completeness I Proof by Contradiction: A ∩ ¬B. (Start with A AND

Consequences of ¬B and break mathematics.) Completeness I Contrapositive Proof: ¬B =⇒ ¬A. (Start with ¬B and Cantor’s Theorem derive ¬A.) Epilogue Remark 19 Be careful when you negate a statements containing “none,” “at least one,” “for all,” and “there exists.”

I The negation of “for all” is “there exists.”

I The negation of “there exists” is “for all.”

I The negation of “at least one” is “none.”

I The negation of “none” is “at least one.” Remember to use De Morgan’s Laws when negating statements about sets. Chapter 1: The Real Numbers Proofs by Contradiction PWhite

Discussion

Some Preliminaries Theorem 20 √ The Axiom of If m is irrational, then 1 + m is irrational. Completeness Proof: Consequences of Completeness

Cantor’s Theorem

Epilogue Chapter 1: The Real Numbers Example PWhite

Discussion

Some Example 21 Preliminaries Two real numbers a and b are equal if and only if for The Axiom of Completeness every  > 0, it follows that |a − b| < . Consequences of Completeness Proof:

Cantor’s Theorem

Epilogue Chapter 1: The Real Numbers Induction PWhite Fundamental Principle of Induction Discussion Let S ⊆ N. If Some Preliminaries 1.1 ∈ S The Axiom of 2. n ∈ S =⇒ n + 1 ∈ S Completeness Then S = . Consequences of N Completeness Cantor’s Theorem The Typical Usage Epilogue Given a statement P(n), where n ≥ n0 and n0 ∈ N ∪ {0}. 1. Base Case: P(n0) is true. 2. Induction Hypothesis: Assume P(k) is true for some k ≥ n0. 3. Inductive Step: Show that P(k) =⇒ P(k + 1). Remark 22 When using induction, I want to see all three steps clearly labeled. Chapter 1: The Real Numbers Induction Example PWhite

Discussion

Some Preliminaries Example 23 The Axiom of n Completeness Prove that for every n ≥ 4, n! > 2 .

Consequences of Completeness

Cantor’s Theorem

Epilogue Chapter 1: The Real Numbers Example PWhite

Discussion Example 24

Some Prove that for any n ≥ 2, the number Preliminaries

The Axiom of r Completeness q √ 1 + 1 + 1 + ··· Consequences of Completeness Cantor’s Theorem is always irrational. (There are n radicals and n 1’s.) Epilogue Chapter 1: The Real Numbers Homework PWhite

Discussion

Some Preliminaries Pages 11 – 13 The Axiom of Problems: 1.2.2, 1.2.3, 1.2.5, 1.2.8, 1.2.10, 1.2.11 Completeness

Consequences of Completeness

Cantor’s Theorem

Epilogue Chapter 1: The Real Numbers Overview PWhite

Discussion

Some √ Preliminaries Discussion: The Irrationality of 2 The Axiom of Completeness Consequences of Some Preliminaries Completeness

Cantor’s Theorem

Epilogue The Axiom of Completeness

Consequences of Completeness

Cantor’s Theorem

Epilogue Chapter 1: The Real Numbers What is R? PWhite I The real numbers contains the rational numbers. Discussion I It “fills in” all the gaps of the rational numbers. Some Preliminaries I It is a field. The Axiom of Completeness I It is ordered – that is, if a 6= b, then either a < b or

Consequences of a > b. Completeness I The real numbers satisfies the Axiom of Cantor’s Theorem Completeness. Epilogue

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

Remark 25 The Axiom of Completeness is an axiom. It is a fundamental building block. Thus we assume that it is true and can’t actually prove that it is true. Chapter 1: The Real Numbers Least Upper and Greatest Lower Bounds PWhite

Discussion Definition 26

Some I A set A ⊆ is bounded above if there exists a Preliminaries R

The Axiom of number b ∈ R such that a ≤ b for all a ∈ R. The Completeness number b is called an upper bound for A. Consequences of Completeness I A set A ⊆ R is bounded below if there exists a Cantor’s Theorem number l ∈ R such that l ≤ a for all a ∈ R. The Epilogue number l is called an lower bound for A. I The supremum or least upper bound for a set A ⊆ R, s = sup(A), satisfies the following 1. s is an upper bound for A. 2. if b is any upper bound for A, then s ≤ b. I The infimum or greatest lower bound for a set A ⊆ R, g = inf(A), satisfies the following 1. g is a lower bound for A. 2. if b is any lower bound for A, then b ≤ g. Chapter 1: The Real Numbers Maximum and Minimum PWhite

Discussion Definition 27

Some I The real number a is a maximum of the set A if a Preliminaries 0 0

The Axiom of is an element of A and a0 ≥ a for all a ∈ A. Completeness I The number a1 is a minimum of the set A if a1 is an Consequences of Completeness element of A and a1 ≤ b for all b ∈ A. Cantor’s Theorem Epilogue Remark 28

I Supremum and infimum (when they exist) are unique. (Easy proof).

I A set can have a supremum and yet have no maximum.

I A set can have an infimum and yet have no minimum. Chapter 1: The Real Numbers Supremum and Infimum PWhite

Discussion Example 29

Some What are the (intuitive) infimum and supremum, as well Preliminaries as maximum and minimum for the following sets? The Axiom of Completeness 1. (0, 1] Consequences of Completeness 2. {1, 2, 3,...} Cantor’s Theorem 3. Q Epilogue  1 4. n | n ∈ N Example 30 Let S = {r ∈ Q | r 2 < 2}. Answer the following questions about S. 1. Explain why sup(S) exists. 2. Is sup(S) ∈ Q? 3. Does inf(S) exist? Is it in Q? 4. Is there a maximum? Minimum? Chapter 1: The Real Numbers Tiny Theory PWhite

Discussion Lemma 31 Some Preliminaries Assume s ∈ R is an upper bound for a set A ⊆ R. Then, The Axiom of s = sup(A) if and only if, for every choice of ε > 0, there Completeness exists an element a ∈ A satisfying s − ε < a. Consequences of Completeness Proof: Cantor’s Theorem

Epilogue Chapter 1: The Real Numbers It all holds for Infimums PWhite

Discussion Remark 32

Some I The Axiom of Completeness does not explicitly say Preliminaries

The Axiom of anything about infimums and their existence. Completeness I However, infimums exist for any set that is bounded Consequences of Completeness below. Cantor’s Theorem I So, any result about supremums has an analogous Epilogue result for infimums. Chapter 1: The Real Numbers Homework PWhite

Discussion

Some Preliminaries Pages: 17 – 18 The Axiom of Problems: 1.3.3, 1.3.4, 1.3.5, 1.3.6, 1.3.7, 1.3.9 Completeness

Consequences of Completeness

Cantor’s Theorem

Epilogue Chapter 1: The Real Numbers Overview PWhite

Discussion

Some √ Preliminaries Discussion: The Irrationality of 2 The Axiom of Completeness Consequences of Some Preliminaries Completeness

Cantor’s Theorem

Epilogue The Axiom of Completeness

Consequences of Completeness

Cantor’s Theorem

Epilogue Chapter 1: The Real Numbers Application of the Axiom of Choice PWhite

Discussion Theorem 33 (Nested Interval Property) Some For each n ∈ N, assume we are given a closed interval Preliminaries In = [an, bn]. Assume also that The Axiom of Completeness Consequences of I1 ⊇ I2 ⊇ I3 ⊇ · · · ⊇ In ⊇ · · · . Completeness Cantor’s Theorem Then, ∞ Epilogue \ In 6= ∅. n=1

Proof:

I Chapter 1: The Real Numbers The Archimedean Property PWhite

Discussion Theorem 34 (The Archimedean Property)

Some Preliminaries I Given any number x ∈ R, there exists an n ∈ N The Axiom of satisfying n > x. Completeness

Consequences of I Given any real number y > 0, there exists an n ∈ N Completeness 1 satisfying < y. Cantor’s Theorem n Epilogue Chapter 1: The Real Numbers Density of Q in R PWhite

Discussion Some Theorem 35 (Density of Q in R) Preliminaries For every two real numbers a and b with a < b, there The Axiom of Completeness exists a rational number r satisfying a < r < b. Consequences of Completeness Proof:

Cantor’s Theorem

Epilogue Chapter 1: The Real Numbers Density of the Irrational Numbers in R PWhite

Discussion Corollary 36

Some Given any two real numbers a < b, there exists an Preliminaries irrational number t satisfying a < t < b. The Axiom of Completeness Remark 37 Consequences of Completeness I We have now shown that for any two rational Cantor’s Theorem numbers, there exists an between Epilogue them. (The previous corollary).

I We have also shown that for any two irrational numbers, there exists a rational number between them. (The previous theorem).

I These two statements will be disturbing when we talk about the “size” of the rational and irrational numbers. Chapter 1: The Real Numbers The Existence of Square Roots PWhite

Discussion

Some Preliminaries Theorem 38 2 The Axiom of There exists a real number α ∈ R satisfying α = 2. Completeness Proof: Consequences of Completeness

Cantor’s Theorem

Epilogue Chapter 1: The Real Numbers Cardinality of Sets PWhite

Discussion Definition 39

Some The cardinality of a set A, |A| = card(A), is the size of a Preliminaries set. The Axiom of Completeness Definition 40 Consequences of Completeness Let f : A 7→ B be a function. Cantor’s Theorem I f is one-to-one, 1-1, if a1 6= a2 in A implies that Epilogue f (a1) 6= f (a2) in B. I f is onto if, given any b ∈ B, there exits a ∈ A such that f (a) = b.

I The sets A and B have the same cardinality, A ∼ B, if f is 1-1 and onto. Chapter 1: The Real Numbers Example PWhite

Discussion Example 41

Some Preliminaries I Let E be the set of even numbers. Show that E ∼ N The Axiom of using f (n) = 2n. Completeness ( n−1 Consequences of 2 , n odd Completeness I Show that N ∼ Z using f (n) = − n , n even. Cantor’s Theorem 2

Epilogue x I Show that f :(−1, 1) 7→ given by f (x) = is R x2 − 1 1-1 and onto. Hence, (−1, 1) ∼ R. Chapter 1: The Real Numbers Equivalence Relations PWhite

Discussion Definition 42

Some A , R, is an if Preliminaries 1. ARA. The Axiom of Completeness 2. ARB =⇒ BRA. Consequences of Completeness 3. ARB and BRC =⇒ ARC. Cantor’s Theorem Epilogue Example 43 Show that the countable relation ∼ is an equivalence relation between sets. Chapter 1: The Real Numbers Countable Sets PWhite

Discussion Definition 44

Some I A set A is countable if ∼ A. Preliminaries N The Axiom of I An infinite set that is not countable is called an Completeness uncountable set. Consequences of Completeness Cantor’s Theorem Remark 45 Epilogue The sets E = even numbers, O = odd numbers, N (duh!) and Z are countable. Theorem 46 1. The set Q is countable. 2. The set R is uncountable. Proof: Chapter 1: The Real Numbers Remark PWhite

Discussion Theorem 47

Some If A ⊆ B and B is countable, then A is either countable, Preliminaries finite, or empty. The Axiom of Completeness Theorem 48 Consequences of Completeness 1. If A1, A2, A3,..., Am are each countable sets, then Cantor’s Theorem Sm the union n=1 An is countable. Epilogue S∞ 2. If An is countable for all n ∈ N, then n=1 An is countable.

Remark 49

I A previous theorem states that Q is countable and R is uncountable. Hence, the irrationals must be uncountable and have a larger cardinality than Q. I But, for any two irrationals, there is a rational between them. The infinity concept is unintuitive. Chapter 1: The Real Numbers Homework PWhite

Discussion

Some Preliminaries Pages: 27 – 29 The Axiom of Problems: 1.4.2, 1.4.4, 1.4.5, 1.4.8 Completeness

Consequences of Completeness

Cantor’s Theorem

Epilogue Chapter 1: The Real Numbers Overview PWhite

Discussion

Some √ Preliminaries Discussion: The Irrationality of 2 The Axiom of Completeness Consequences of Some Preliminaries Completeness

Cantor’s Theorem

Epilogue The Axiom of Completeness

Consequences of Completeness

Cantor’s Theorem

Epilogue Chapter 1: The Real Numbers Cantor’s Diagonalization Method PWhite

Discussion Theorem 50 Some Preliminaries The open interval (0, 1) = {x ∈ R | 0 < x < 1} is The Axiom of uncountable. Completeness

Consequences of Proof: Note: The proof uses Cantor’s Diagonalization Completeness Method. Cantor’s Theorem

Epilogue Chapter 1: The Real Numbers Power Sets PWhite

Discussion Definition 51

Some Given a set A, the of A, P(A), is the set of all Preliminaries of A. The Axiom of Completeness Example 52 Consequences of Completeness Let A = {a, b, c}. Cantor’s Theorem 1. Find P(A). Epilogue 2. If A has n elements, then |P(A)| = 2n. 3. Notice that it is trivial to construct a 1 − 1 function f : A 7→ P(A). However, it is impossible to find an onto function. Chapter 1: The Real Numbers Cantor on Power Sets PWhite

Discussion

Some Theorem 53 (Cantor’s Theorem) Preliminaries Given any set A, there does not exist a function The Axiom of Completeness f : A 7→ P(A) that is onto. Consequences of Completeness Proof:

Cantor’s Theorem

Epilogue Chapter 1: The Real Numbers Homework PWhite

Discussion

Some Preliminaries Pages: 30 – 32 The Axiom of Problems: 1.5.1, 1.5.4, 1.5.9 Completeness

Consequences of Completeness

Cantor’s Theorem

Epilogue Chapter 1: The Real Numbers Overview PWhite

Discussion

Some √ Preliminaries Discussion: The Irrationality of 2 The Axiom of Completeness Consequences of Some Preliminaries Completeness

Cantor’s Theorem

Epilogue The Axiom of Completeness

Consequences of Completeness

Cantor’s Theorem

Epilogue Chapter 1: The Real Numbers Something Odd PWhite I Let ℵ0 = | |. This is the first cardinal number (the Discussion N smallest infinity). Some Preliminaries I Let c = |R|. This is the second cardinal number. The Axiom of Completeness I We’ve shown that ℵ0 < c. Consequences of I The Question: Does there exist A ⊂ R such that Completeness ℵ0 < |A| < c? Cantor’s Theorem I Cantor’s Continuum Hypothesis: There is no A ⊂ R Epilogue such that ℵ0 < |A| < c. I Kurt Gödel: Using the agreed upon axioms of , the continuum hypothesis could not be disproven. I Paul Cohen: Using the same axioms, the continuum hypothesis could not be proven. I So, we are free to accept or not the continuum hypothesis without developing any logical contradictions.