Chapter 1

Review Material: Sets,Functions andMathematical Induction

This chapter contains material students are expected to know. Except for someparts of a few sections, this material will not be covered thoroughly in class.However, if you nd that you do not remember parts of this material, it is yourresponsibility to review it. One way to test your knowledge of this material is totry the homework at the end of each section. If you nd you struggle doing thehomework in a section it means you need to review the material in that section.Here is the material that will be discussed in class from this section:

Review of Sets: notation, important results and some homework problems. Functions: direct and inverse image of a set. Proof by induction. Set of real numbers: Brief introduction, absolute value.

1.1 Review of Sets

This section is only intended as a quick review of material the reader needsto know for these notes. For a more thorough treatment of sets, the reader isinvited to read books on set theory.

Denition 1 (Set) A set is a well-dened, unordered collection of distinctobjects. Well-dened means that given an object and a set, we can tell if theobject is in the set. The objects in a set are called its elements. If x is anelement of a set A, we write x 2 A. This is read "x is an element of A".

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2CHAPTER 1. REVIEWMATERIAL: SETS, FUNCTIONS ANDMATHEMATICAL INDUCTION

A set can be nite or innite. Sets are usually named using a capital lettersuch as A, B, ... The words sets, collection, family are all synonymous. Theorder in which the elements are listed is not relevant. In other words, two setsare equal if and only if they contain the same elements, regardless of the orderin which they appear.

1.1.1 Examples and Notation

There are dierent ways of representing a set.

1. We can list all the elements of the set, or, we list enough elements until apattern is established, then we add .... In both cases, the elements aresurrounded by curly brackets.

Example 2 A = f1; 2; 4g represents the set consisting of the numbers 1,2, and 4.

Example 3 B = f1; 2; 3; :::g represents the set of natural numbers.

Example 4 P = ff1; 2g ; f2; 3gg. The elements of a set can be sets them-selves.

2. We give a rule or a condition to be satised in order to belong to a set. IfP (x) denotes the condition to be satised, we can dene a set by writingfx : P (x)g. This means the set of x0s such that P (x) is true. Since thisis real analysis, we will assume that the x0s come from R. If this is notthe case, we must specify the set they come from. If they come from a setA, then we write fx 2 A : P (x)g.

Example 5 A = fx 2 N : x is eveng is the same as f2; 4; 6; :::g

Remark 6 In the above denition, : is read "such that".

Example 7 C = fx 2 R : 2 < x < 5g

Example 8 Q=nmn

: m 2 Z, n 2 Z and n 6= 0o

1.1.2 Special Symbols

There are some special symbols associated with sets. Some of these symbolsinclude:

2 means is a member of. When we write 2 2 A, we mean that theelement 2 is a member of the set called A.

=2 means is not a member of.

1.1. REVIEW OF SETS 3

means "is included in but not equal to" or is a proper subset of.It is used between two sets. When we write A B, we mean that A iscontained in B, the two are not equal.

means "is included in and could be equal to" or is a subset of; it isused between two sets. When we write A B, it is understood that A iscontained in B and could be equal to B:

? denotes the empty set. It is also called the void set or the null set. If A is a set, we denote jAj the cardinality of A that is the number ofelements of A. If A has n distinct elements, then jAj = n and A is said tobe nite. If A is not nite, it is said to be innite.

Remark 9 Let A and B be two sets.

1. To show A B, one must show that if we pick an arbitrary element x inA then x is also in B.

2. To show A B one must show that A B and that B has at least oneelement which is not in A.

3. It should be clear that ? A for every set A.

Some sets which are used often have a special name.

N = f1; 2; 3; :::g represents the set of natural numbers or positive in-tegers.

Z = f:::;2;1; 0; 1; 2; :::g represents the set of integers.

Q =nmn

: m;n 2 Z and n 6= 0orepresents the set of rational numbers.

R represents the set of real numbers. If A is any set, then the power set of A, denoted P (A), is the set of allsubsets of A (including ? and A). For example if A = f1; 2; 3g then

P (A) = f?; f1g ; f2g ; f3g ; f1; 2g ; f1; 3g ; f2; 3g ; f1; 2; 3gg

It can be shown that if A has n elements, then P (A) has 2n elements thatis jP (A)j = 2n.

1.1.3 Operations on Sets

There are four operations on sets you should know about. They are [ (union),\ (intersection), (or n) (dierence or relative complement) and (Cartesianproduct). These operations are dened below. Let us assume we have two setsA and B.

4CHAPTER 1. REVIEWMATERIAL: SETS, FUNCTIONS ANDMATHEMATICAL INDUCTION

Union

The union of A and B, denoted A [B is dened by:A[B = fx : x 2 A or x 2 Bg i.e. it consists of the elements which are either

in A or in B.

Intersection

The intersection of A and B, denoted A \B is dened by:A \B = fx : x 2 A and x 2 Bg i.e. it consists of the elements which are in

both A and B.

Relative Complement

The relative complement of B in A, denoted A nB or AB (we will use A nBas AB also means fa b : a 2 A and b 2 Bg) is dened by:

A n B = fx : x 2 A and x =2 Bg i.e. it consists of the elements of A whichare not also in B.Sometimes, when all the sets we are working with are subsets of a xed set

we will call X, then instead of writing X n A or X A, we write AC . This setis called the complement of A. It is understood that it is the complement ofA in X. For example, in these notes, R is the set where all our elements comefrom, unless specied otherwise. So, if we write AC , it will mean RA.If A and B are two sets, A nB 6= B nA in general.

Cartesian Product

The Cartesian product of A and B, denoted AB is dened by:AB = f(x; y) : x 2 A and y 2 Bg. This is called the Cartesian product of

A and B. It is the set of all ordered pairs whose rst element comes from Aand second element from B.Because it is a set of ordered pairs, if A and B are dierent sets, A B 6=

B A.Remark 10 The Cartesian product of R with itself, R R is denoted R2.

Figures 1.1, 1.2 and 1.3 illustrate the rst three operations.

Example 11 Given A = f0; 1; 2; 3; 4; 5; 6g and B = f5; 6; 7; 8; 9g, then we have:A [B = f0; 1; 2; 3; 4; 5; 6; 7; 8; 9gA \B = f5; 6gA nB = f0; 1; 2; 3; 4gB nA = f7; 8; 9g

Example 12 Given A = fa; bg and B = f1; 2g we have:AB = f(a; 1) ; (a; 2) ; (b; 1) ; (b; 2)gB A = f(1; a) ; (1; b) ; (2; a) ; (2; b)g

1.1. REVIEW OF SETS 5

Figure 1.1: Union of Sets: A [B

Figure 1.2: Intersection of Sets: A \B

6CHAPTER 1. REVIEWMATERIAL: SETS, FUNCTIONS ANDMATHEMATICAL INDUCTION

Figure 1.3: Dierence of Sets: AB

Remark 13 From the two examples above, you notice that, in general, AnB 6=B nA and AB 6= B A.

1.1.4 Some Results About Sets

Let us make a few remarks before we state and prove some important results.

Remark 14 The standard technique to prove that two sets A and B are equal(that is A = B) is to prove that A B and B A.

Remark 15 Often, when proving some result about the elements of a set A,we start by picking an arbitrary element of that set by using a statement like"Let x 2 A....". However, this statement makes sense only if A has elements,in other words A 6= ?. If we do not know that fact, we will have to consider thespecial case A = ? as part of the proof.

Theorem 16 Let A; B; and C be three sets. Then, the following properties aresatised:

1. ? A2. A \A = A, A [A = A3. A \? = ?, A [? = A4. Commutative law. A [B = B [A and A \B = B \A5. Associative law. A[(B [ C) = (A [B)[C and A\(B \ C) = (A \B)\C

1.1. REVIEW OF SETS 7

6. Distributive laws. A [ (B \ C) = (A [B) \ (A [ C) and A \ (B [ C) =(A \B) [ (A \ C)

7. If in addition, A B then A [B = B and A \B = AProof. We only prove a few of the statements. For the other ones, see problemsat the end of the section.

1. For this proof, it is easier to consider what it would take for this statementto be false. There would have to be an element of ? which is not in A. Butthis cannot happen since there are no elelements in ?. So the statementcannot be false, hence it is true.

2. We only prove A\A = A. Though this statement seems pretty obvious, weneed to prove it. This can only be done by using "irrefutable" arguments.One way to show two sets are equal is to show we have inclusion bothways. So, we show A \A A and A A \A. A\A A. If A\A = ? then the result holds since the empty set is asubset of every set. If A\A 6= ? then we pick an arbitrary x 2 A\Aand show that x 2 A. If x 2 A \A =) x 2 A and x 2 A =) x 2 A.

A A \ A. If A = ? then the result holds since the empty set isa subset of every set. If A 6= ? then we pick an arbitrary x 2 Aand show that x 2 A \ A. If x 2 A =) x 2 A =) x 2 A andx 2 A =) x 2 A \A.

Denition 17 (Disjoint Sets) Two nonempty sets A and B are said to bedisjoint if they do not intersect that is if A \B = ?.Remark 18 It should be clear to the reader that if A = ? or B = ? thenA \B = ?.Theorem 19 A \B and A nB are disjoint sets. Furthermore, A = (A \B) [(A nB).Proof. We rst prove the sets are disjoint. We then prove A = (A \B) [(A nB), by showing that A (A \B) [ (A nB) and (A \B) [ (A nB) A. Proof that the sets are disjoint. The result seems obvious if we look atFigure 1.3. However, this is just a specic case. We need to establish thisfact in general. We show the sets are disjoint by showing an element xcannot be in both sets at the same time. If x 2 A\B, then it is in both Aand B. Hence, it cannot be in A nB. Similarly, if x 2 A nB, then x =2 Bhence it cannot be in A \B.

Proof that A (A \B) [ (A nB). If A = ?, the result follows since ?is a subset of any set. If A 6= ?, the we pick x 2 A. Either x 2 B inwhich case x 2 A \ B, or x =2 B in which case x 2 A n B. It follows thatx 2 (A \B) [ (A nB) hence A (A \B) [ (A nB).

8CHAPTER 1. REVIEWMATERIAL: SETS, FUNCTIONS ANDMATHEMATICAL INDUCTION

Proof that (A \B) [ (A nB) A. If (A \B) [ (A nB) = ?, the resultfollows since ? is a subset of any set. If (A \B) [ (A nB) 6= ? then letx 2 (A \B) [ (A nB). Either x 2 A \ B or x 2 A n B. In both cases.x 2 A hence (A \B) [ (A nB) A.

Theorem 20 (De Morgans Laws) Let A, B, and C be sets. Then the fol-lowing holds:

1. A n (B \ C) = (A nB) [ (A n C)2. A n (B [ C) = (A nB) \ (A n C)

Proof. We only prove the rst part. The second one is left as an exercise.First, we show that A n (B \ C) (A nB) [ (A n C). Let x 2 A n (B \ C).Then, x 2 A and x =2 B \ C. So, x 2 A and either x =2 B or x =2 C . If x =2 Bthen x 2 A n B since x 2 A. Similarly, if x =2 C then x 2 A n C. Therefore,either x 2 A nB or x 2 A n C. Hence, x 2 (A nB) [ (A n C).Next, we show that (A nB)[ (A n C) A n (B \ C). Let x 2 (A nB)[ (A n C).Either x 2 AnB or x 2 AnC. So, either x 2 A and x =2 B or x 2 A and x =2 C.In both cases, x 2 A and either x =2 B or x =2 C. So, x 2 A and x =2 B \C thatis x 2 A n (B \ C).

Remark 21 If we assume that both A and B are subsets of a xed set X andwe wish to express the relative complement with respect to X, then theorem 20can be written

1. (A \B)c = Ac [Bc

2. (A [B)c = Ac \Bc

1.1.5 Exercises

1. Prove Theorem 16.

2. Prove the second part of Theorem 20.

3. Let A and B be two non-empty sets.

(a) Prove that A \B A(b) Prove that A A [B

4. Let D denote the set of elements which either belong to A or to B butnot to both. Prove that D = (A nB) [ (B nA). This set is often calledthe symmetric dierence of A and B.

5. Show that the set D dened in the previous problem is also given byD = (A [B) n (A \B)

1.1. REVIEW OF SETS 9

6. Read about Richard Dedekind and his contribution to the construction ofthe set of real numbers.

7. Do the same for Georg Cantor.

8. Let

A = f1; 0; 1; 2gB = f2; 3gC = f2; 0; 1; 5g

(a) Find the following:

A [B B [ C A \B B \ C A \ C A \ (B [ C) A nB C nB A n (B [ C)

(b) Find the following:

AB C B (AB) \ (C B) (A \ C)B

(c) On the basis of your answer in (b), what might you conjecture about(A \ C)B for arbitrary sets A;B;C?

9. Let

A = fx 2 R : 1 x 5gB = fx 2 R : 0 x 3gC = fx 2 R : 2 x 4g

(a) Find each of the following:

A \B A \ Z B \ C A [B B [ C

(b) Find each of the following:

10CHAPTER 1. REVIEWMATERIAL: SETS, FUNCTIONS ANDMATHEMATICAL INDUCTION

AB A C (AB) [ (A C) A (B [ C)

(c) Sketch the sets (AB) [ (A C) and A (B [ C)(d) On the basis of your answer in (c), what might you conjecture about

A (B [ C) for arbitrary sets A;B;C?10. If A is a subset of X, prove that:

(a) A [Ac = X(b) A \Ac = ?(c) (Ac)c = A

11. Verify that a set with three elements has eight subsets.

12. If A and B are subsets of a set X, prove that A nB = A \Bc

13. If A and B are any sets prove that A nB and A\B are disjoint and thatA = (A nB) [ (A \B)

14. Prove that A (B1 [B2) = (AB1) [ (AB2). Prove the same resultfor intersection.

15. Suppose that A;C are subsets of X and B;D are subsets of Y . Prove that

(AB) \ (C D) = (A \ C) (B \D)

16. What is jP (?)j?17. Let A and B be sets. Prove that A \B A [B.18. Let A, E and F be three sets.

(a) Prove that if both E A and F A then E \ F A.(b) Prove hat if A E and A F then A E [ F .(c) Prove hat if A E and A F then A E \ F .(d) Prove that if both E A and F A then E [ F A.(e) Prove that if E F then A \ E A \ F .(f) Prove that if E F then A [ E A [ F .

19. Let A, B and C be sets. Determine if the statements below are true orfalse. If you think they are true, prove them. If you think they are wrong,either explain why or give a counterexample.

(a) If A B \ C then A B and A C.(b) If A B [ C then A B or A C.

Review Material: Sets, Functions and Mathematical InductionReview of SetsExamples and NotationSpecial SymbolsOperations on SetsSome Results About SetsExercises

FunctionsWhat is a Function?Operations on FunctionsInjections and SurjectionsInverse FunctionsDirect Image and Inverse Image of a SetAdditional Properties of a FunctionExercises

Induction and Other Proof TechniquesReview of Proof techniques Other than InductionMathematical InductionExercises

The set of Real Numbers: Quick Definition and Basic PropertiesIntuitive Definition of the Real NumbersAlgebraic Properties of R Important Basic Properties of Real NumbersExercises

Order Axioms and Order Properties of R Order Axioms for R and DefinitionsProperties of the OrderExercises

Metric Properties of R - Absolute ValueDefinition and PropertiesEquations and Inequalities Involving Absolute ValuesNotion of a Metric SpaceExercises

Important Properties of R The Completeness Property of R Bounded SetsThe Axiom of CompletenessIntervalsExercises

Some Consequences of the Completeness AxiomArchimedean PropertyDenseness of Q in R Exercises

Boundedness of FunctionsSuprema and Infima of a FunctionExercises

Sequences, Sequences of SetsSequencesSequences of Sets and Indexed Families of SetsExercises

CardinalityIntroductionFinite and Infinite SetsCountable and Uncountable SetsExercises

The Extended Real Number SystemIntroductionArithmetic in the Extended Real Number SystemSuprema and InfimaExercises

Topology of the Real LineInterior Points and NeighborhoodsMain DefinitionsPropertiesTechniques to RememberImportant Facts to Know and RememberExercises

Open and Closed SetsMain DefinitionsPropertiesClosure of a SetTechniques to RememberImportant Facts to Know and RememberExercises

Limit PointsMain DefinitionsPropertiesImportant Facts to RememberExercises

Compact SetsExercises

Topology of the Real Line: Review Questions

Sequences and Limit of SequencesSequences: Basic DefinitionsLimit of a Sequence: DefinitionsThe Concepts of Eventually and FrequentlyDefinitions of LimitsExamplesExercises

Limit of a Sequence: TheoremsLimit PropertiesLimit LawsMore Theorems on LimitsExercises

Monotone Sequences and Cauchy SequencesMonotone SequencesCauchy SequencesExercises

SubsequencesExercises

Limits of a FunctionDefinitions and ExamplesOverviewLimit at a finite pointComputing Limits Using the definitions: ExamplesExercises

Limit TheoremsOperations with LimitsElementary TheoremsRelationship Between the Limit of a Function and the Limit of a SequenceExercises

Monotone FunctionsExercises

ContinuityPreliminary NotionsDefinitionsSome Properties of Continuous FunctionsDiscontinuitiesExercises

Deeper Properties of Continuous FunctionsIntermediate Value Theorem and ConsequencesTopological Interpretation of ContinuityExercises

Uniform ContinuityExercises

DifferentiationA Quick Review from Differential CalculusDefinitionsThe Derivative as a Function, Higher Order DerivativesRules of DifferentiationExercises

Important TheoremsLocal Extrema and Fermat's TheoremIntermediate Value Property of the DerivativeMean Value TheoremsExercises

Riemann IntegrationIntroductionDefinition of the Riemann-Darboux IntegralNotion of a PartitionDarboux SumsRiemann-Darboux IntegralExercises

The Riemann IntegralRiemann SumsRiemann IntegralRelationship Between the Riemann-Darboux and the Riemann IntegralsExercises

Properties of the Riemann IntegralExercises

Relationship Between Integration and DifferentiationThe Fundamental Theorem of CalculusThe Mean Value Theorem for Integrals and Other Important TheoremsExercises

Improper IntegralsImproper Integrals of the First KindImproper Integrals of the Second KindExercises

Infinite SeriesConvergence and DivergenceGeneral ConceptsGeometric SeriesMore General Convergence TestsTesting Series With Positive TermsExercises

Solution to the Homework Problems

Solutions to Problems Which Appear in the TextSolutions to Problems from Chapter 1Solutions to Problems from Section 1.1.5 on Page 8Solutions to Problems from Section 1.2.7 on Page 20Solutions to Problems from Section 1.3.3 on Page 27Solutions to Problems from Section 1.4.4 on Page 34Solutions to Problems from Section 1.5.3 on Page 37Solutions to Problems from Section 1.6.4 on Page 42

Solutions to Problems from Chapter 2Solutions to Problems from Section 2.1.4 on Page 52Solutions to Problems from Section 2.2.3 on page 58Solutions to Problems from Section 2.3.2 on page 61Solutions to Problems from Section 2.4.3 on page 67Solutions to Problems from Section 2.5.4 on page 75Solutions to Problems from Section 2.6.4 on page 77

Solutions to Problems from Chapter 3Solutions to Problems from Section 3.1.5 on Page 82Solutions to Problems from Section 3.2.6 on Page 89Solutions to Problems from Section 3.3.4 on Page 94Solutions to Problems from Section 3.4.1 on Page 95Solutions to Problems from Section 3.5 on Page 96

Solutions to Problems from Chapter 4Solutions to Problems from Section 4.2.4 on page 109Solutions to Problems from Section 4.3.4 on page 123Solutions to Problems from Section 4.4.3 on page 132Solutions to Problems from Section 4.5.1 on page 134

Solutions to Problems from Chapter 5Solutions to Problems from Section 5.1.4 on page 145Solutions to Problems from Section 5.2.4 on page 151Solutions to Problems from Section 5.3.1 on page 152

Solutions to Problems from Chapter 6Solutions to Problems from Section 6.1.4 on Page 163Solutions to Problems from Section 6.2.3 on Page 168Solutions to Problems from Section 6.3.1 on Page 172

Solutions to Problems from Chapter 7Solutions to Problems from Section 7.1.4 on Page 178Solutions to Problems from Section 7.2.4 on Page 183

Solutions to Problems from Chapter 8Solutions to Problems from Section 8.2.4 on Page 197Solutions to Problems from Section 8.3.4 on Page 202Solutions to Problems from Section 8.4.1 on Page 205Solutions to Problems from Section 8.5.3 on Page 211Solutions to Problems from Section 8.6.3 on Page 219

Bibliography