Mathematical Analysis

Real Analysis lecture photos:


Sofia Kovalevskaya 1850 – 1891


Cathleen Morawetz 1923 -2017

I find that I may have emphasized the need to escape from the devils of mathematics to embark on the pleasures of the real world. But it works both ways, and sometimes the devils of the real world drive one into the pleasures of studying mathematics.

Mathematicians on Creativity by Peter Borwein

Mathematical Analysis

Below is a set of notes on Mathematical Analysis

Introductory CHAPTER:  Mathematical Logic, Proof, and Sets

Section A: Joy of Sets

Section A Introduction to Joy of Sets video

Section A Exercise Ia Complete Solutions to Exercise Ia
Section B: Subsets (Video)

Subsets notes

Section B: Power Set Video

Power set notes

Proving results about Subsets (Video)

Proving results Subsets

SECTION B Subsets Exercise I  (b) Complete Solutions to Exercise I(b)
Section C: Proposition Logic

Section I(c) Exercise I(c)
Section D: Algebra of Propositions

Video of lecture

Introductory: Algebra of Propositions Section D

Section I(d) Exercise I(d) Complete solutions to Exercise I(d)
Section E: Implication

Video of lecture

Introductory Section E Implication

Introductory Section E2 Tautology

Tautology and Contradiction part I

Tautology and Contradiction part II

Sections E4 and E5: Converse and Contrapositive

Section I(e)

On the following test enter your truth values in numerical order.

Test on Truth Tables
Exercise I(e) Complete solutions to Exercise I(e)
Section F: Introduction to Proof

Section F1: If and only if

Section F2: Introduction to Proof

Section I(f) Exercise I(f) Complete solutions to Exercise I(f)
Section G: Proofs

Section G2: Proof by Contrapositive

Section G3: Without Loss of Generality

SECTION I(g) PROOFS Exercise I(g)
Section H: Proof by Contradiction

Section H: Proof by Contradiction

H1: Introduction to Proof by Contradiction

Section H2: Examples of Proof by Contradiction

SECTION I(h) PROOF  BY CONTRADICTION Exercise I(h) Complete Solutions to Exercise I(h)
Section I: Principle of Mathematical Induction Section I(i) Exercise I(i) Complete Solutions to Exercise I(i)

Chapter 1 : Logic and Proof

Section A:

Propositional Logic

userfiles/Section A Propositional Logic(1).pdf userfiles/Exercise 1(a)(1).pdf
Section B:

Algebra of Propositions

userfiles/Section B Algebra of Propositions.pdf userfiles/Exercise 1b.pdf userfiles/Complete solutions to Exercise 1b.pdf
 Section C:


 userfiles/Section C Implication.pdf  userfiles/Exercise 1(c).pdf  userfiles/Complete solutions to Exercise 1(c)(1).pdf
 Section D:

Introduction to Proof

 userfiles/Section D Introduction to Proof.pdf  userfiles/Exercise 1d(1).pdf  userfiles/Complete solutions to Exercise 1d(1).pdf
 Section E:


 userfiles/SECTION E PROOFs.pdf  userfiles/Exercise 1.pdf
 Section F:

Proofs by Contradiction

 userfiles/SECTION F PROOF BY CONTRADICTION.pdf  userfiles/Exercise 1F(1).pdf  userfiles/Complete Solutions to Exercise 1f(1).pdf
 Section G:

Principle of Mathematical Induction

 userfiles/Section G Principle Of Mathematical Induction.pdf  userfiles/Exercise 1g.pdf  userfiles/Complete Solutions to Exercise 1g.pdf
 Section H:

Further Mathematical Induction

 userfiles/SECTION H Further Mathematical Induction.pdf  userfiles/Exercise 1h.pdf  userfiles/Complete Solutions to Exercise 1h.pdf

Notes on inequalities

SECTION D Inequalities     Exercise 4d

Video on Inequalities Part I

Inequalities Part II

Notes on Inequalities

Test on Inequalities

SECTION E Modulus Function

Video on Modulus Function

Notes on modulus function

Video on Inequality Properties of the modulus function 

Notes on Inequality Properties of the modulus function

Exercise 4e

Chapter 2 : Infinite Series

Section A:

Introduction to Infinite Series

Video on Introduction to Infinite Series

Video on Examples of Infinite Series

Video on examples of Geometric Series

userfiles/Introduction to Infinite Series(1).pdf Exercise 2a  Solutions 2a
Section B:

Telescoping and Harmonic Series

Video on finding the sum by partial  fractions

Video on Harmonic Series

userfiles/Telescoping and Harmonic Series.pdf userfiles/Exercise 2b(1).pdf  Solutions 2b
 Section C:

Properties of Convergent and Divergent Series

Section 2C  userfiles/Exercise 2c(1).pdf Solutions 2c
 Section D:

Comparison Test and p-series

userfiles/Section 2D Comparison Test and p-series.pdf  userfiles/Exercise 2d.pdf Solutions 2d
 Section E:

Ratio Test

 userfiles/Section 2E Ratio Test.pdf
Exercise 2e
Solutions 2e
 Section F:

Integral Test

 userfiles/Section 2F Integral Test.pdf  userfiles/Exercise 2f.pdf Solutions 2f
 Section G:

Alternating Series

 userfiles/Section 2G Alternating Series.pdf  userfiles/Exercise 2g.pdf  Solutions 2g
 Section H to be added later.

Section I:

Power Series

 userfiles/Section 2I Power Series.pdf  userfiles/Exercise 2i.pdf

Notes on Pointwise and Uniform Convergence

Sequences of Functions

Pointwise Convergence

 Section B Uniform Convergence
Section C

Section D

Taylor Series and Largest Error

Chapter 3 : Functions

Section A

Introduction to Functions

Videos on

Defintion of a function

Notes on Defn of a function

Range of a function part I

Range of a function part II

Notes on Range of a function

Chapter 3A Functions userfiles/Exercise 3a(1).pdf
Section B:

Injective and Surjective

Videos on

Injective Functions

Notes on Injective functions

Surjective functions

Notes on Surjective functions

SECTION 3B Injective and Surjective functions userfiles/Exercise 3b(1).pdf
Section C:

Inverse Functions


Bijective Functions

Inverse Functions

Section 3c Inverse functions userfiles/Exercise 3c(1).pdf
Section D:

Composite Functions

userfiles/SECTION D Composite Functions.pdf userfiles/Exercise 3d(1).pdf
Section E:

Properties of Composite Functions

userfiles/SECTION E Properties of Composite Functions.pdf userfiles/Exercise 3e(1).pdf
Section F:

Graphs of Functions

userfiles/SECTION F Graphs of Functions.doc userfiles/Exercise 3f.pdf

Chapter 5 : Sequences and Real Numbers

Section A:

Upper and Lower Bounds

userfiles/Section A Upper and Lower Bounds.pdf userfiles/Exercise 5a(1).pdf
Section B:

Properties of Upper and Lower Bounds

userfiles/5B Properties of Upper and Lower Bounds.pdf userfiles/Exercise 5b(1).pdf
Section C:

Limit of a Sequence

Video on

Introduction to limit of a function

Proving limit of a function

userfiles/Section C The Limit of a Sequence.pdf Exercise 5c userfiles/Complete Solutions to Exercise 5c(1).pdf
Section D:

Properties of the Limit of a Sequence

Section D Properties of The Limit of a Sequence userfiles/Exercise 5d.pdf userfiles/Complete Solutions to Exercise 5d.pdf
Section E:

Evaluating Limits

userfiles/Section E Evaluating Limits.pdf userfiles/Exercise 5e.pdf userfiles/Complete Solutions to Exercise 5e.pdf
Section F:

Monotone Sequences

userfiles/Section F Monotone Sequences.pdf userfiles/Exercise 5f.pdf userfiles/Complete solutions to Exercise 5f.pdf
Section G:


userfiles/Section G Subsequences.pdf

Exercise 7H

Complete Solutions to 7h

Exercise 7i

Complete solutions to Exercise 7i.pdf