Pure Mathematics for Beginners: A Rigorous Introduction to Logic, Set Theory, Abstract Algebra, Number Theory, Real Analysis, Topology, Complex Analysis, and Linear Algebra

262 Pages·2018·11.125 MB·English

by  Steve Warner

Legal Notice

This book is copyright 2018 with all rights reserved. It is illegal to copy, distribute, or
create derivative works from this book in whole or in part or to contribute to the
copying, distribution, or creating of derivative works of this book.

For information on bulk purchases and licensing agreements, please email
[e-mail]

Acknowledgements
Thanks to Daniel Dimijian, Scott Jeffreys, Dan Seabold, C.R. Sincock, Pete Terlecky,
Zoran Sunik, and David Wayne for their helpful input during the creation of this book.

CONNECT WITH DR. STEVE WARNER
Pure Mathematics
for Beginners

A Rigorous Introduction to Logic, Set Theory,
Abstract Algebra, Number Theory, Real Analysis,
Topology, Complex Analysis, and Linear Algebra
--------
Dr. Steve Warner

𝑁 (𝐿)
𝜖
𝑁⨀(𝑎) 𝜖
𝛿

𝛿

© 2018, All Rights Reserved

iii
Table of Contents
Introduction 7
For students 7
For instructors 8

Lesson 1 – Logic: Statements and Truth 9
Statements with Words 9
Statements with Symbols 10
Truth Tables 12
Problem Set 1 16
Lesson 2 – Set Theory: Sets and Subsets 19
Describing Sets 19
Subsets 20
Unions and Intersections 24
Problem Set 2 28
Lesson 3 – Abstract Algebra: Semigroups, Monoids, and Groups 30
Binary Operations and Closure 30
Semigroups and Associativity 32
Monoids and Identity 34
Groups and Inverses 34
Problem Set 3 36
Lesson 4 – Number Theory: The Ring of Integers 38
Rings and Distributivity 38
Divisibility 41
Induction 43
Problem Set 4 48
Lesson 5 – Real Analysis: The Complete Ordered Field of Reals 50
Fields 50
Ordered Rings and Fields 52
Why Isn’t ℚ enough? 56
Completeness 58
Problem Set 5 62
Lesson 6 – Topology: The Topology of ℝ 64
Intervals of Real Numbers 64
Operations on Sets 66
Open and Closed Sets 70
Problem Set 6 76
Lesson 7 – Complex Analysis: The Field of Complex Numbers 78
A Limitation of the Reals 78
The Complex Field 78
Absolute Value and Distance 82
Basic Topology of ℂ 85
Problem Set 7 90

iv
Lesson 8 – Linear Algebra: Vector Spaces 93
Vector Spaces Over Fields 93
Subspaces 98
Bases 101
Problem Set 8 105
Lesson 9 – Logic: Logical Arguments 107
Statements and Substatements 107
Logical Equivalence 108
Validity in Sentential Logic 111
Problem Set 9 116
Lesson 10 – Set Theory: Relations and Functions 118
Relations 118
Equivalence Relations and Partitions 121
Orderings 124
Functions 124
Equinumerosity 130
Problem Set 10 135
Lesson 11 – Abstract Algebra: Structures and Homomorphisms 137
Structures and Substructures 137
Homomorphisms 142
Images and Kernels 146
Normal Subgroups and Ring Ideals 147
Problem Set 11 150
Lesson 12 – Number Theory: Primes, GCD, and LCM 152
Prime Numbers 152
The Division Algorithm 155
GCD and LCM 159
Problem Set 12 167
Lesson 13 – Real Analysis: Limits and Continuity 169
Strips and Rectangles 169
Limits and Continuity 172
Equivalent Definitions of Limits and Continuity 175
Basic Examples 177
Limit and Continuity Theorems 181
Limits Involving Infinity 183
One-sided Limits 185
Problem Set 13 186
Lesson 14 – Topology: Spaces and Homeomorphisms 189
Topological Spaces 189
Bases 192
Types of Topological Spaces 197
Continuous Functions and Homeomorphisms 204
Problem Set 14 210

v
Lesson 15 – Complex Analysis: Complex Valued Functions 212
The Unit Circle 212
Exponential Form of a Complex Number 216
Functions of a Complex Variable 218
Limits and Continuity 223
The Reimann Sphere 228
Problem Set 15 230
Lesson 16 – Linear Algebra: Linear Transformations 234
Linear Transformations 234
Matrices 239
The Matrix of a Linear Transformation 242
Images and Kernels 244
Eigenvalues and Eigenvectors 247
Problem Set 16 253

Index 255
About the Author 259
Books by Dr. Steve Warner 260

vi
I N T R O D U C T I O N
PURE MATHEMATICS

This book was written to provide a basic but rigorous introduction to pure mathematics, while exposing
students to a wide range of mathematical topics in logic, set theory, abstract algebra, number theory,
real analysis, topology, complex analysis, and linear algebra.
For students: There are no prerequisites for this book. The content is completely self-contained.
Students with a bit of mathematical knowledge may have an easier time getting through some of the
material, but no such knowledge is necessary to read this book.
More important than mathematical knowledge is “mathematical maturity.” Although there is no single
agreed upon definition of mathematical maturity, one reasonable way to define it is as “one’s ability
to analyze, understand, and communicate mathematics.” A student with a higher level of mathematical
maturity will be able to move through this book more quickly than a student with a lower level of
mathematical maturity.
Whether your level of mathematical maturity is low or high, if you are just starting out in pure
mathematics, then you’re in the right place. If you read this book the “right way,” then your level of
mathematical maturity will continually be increasing. This increased level of mathematical maturity will
not only help you to succeed in advanced math courses, but it will improve your general problem
solving and reasoning skills. This will make it easier to improve your performance in college, in your
professional life, and on standardized tests such as the SAT, ACT, GRE, and GMAT.
So, what is the “right way” to read this book? Simply reading each lesson from end to end without any
further thought and analysis is not the best way to read the book. You will need to put in some effort
to have the best chance of absorbing and retaining the material. When a new theorem is presented,
don’t just jump right to the proof and read it. Think about what the theorem is saying. Try to describe
it in your own words. Do you believe that it is true? If you do believe it, can you give a convincing
argument that it is true? If you do not believe that it is true, try to come up with an example that shows
it is false, and then figure out why your example does not contradict the theorem. Pick up a pen or
pencil. Draw some pictures, come up with your own examples, and try to write your own proof.
You may find that this book goes into more detail than other math books when explaining examples,
discussing concepts, and proving theorems. This was done so that any student can read this book, and
not just students that are naturally gifted in mathematics. So, it is up to you as the student to try to
answer questions before they are answered for you. When a new definition is given, try to think of your
own examples before looking at those presented in the book. And when the book provides an example,
do not just accept that it satisfies the given definition. Convince yourself. Prove it.
Each lesson is followed by a Problem Set. The problems in each Problem Set have been organized into
five levels, Level 1 problems being considered the easiest, and Level 5 problems being considered the
most difficult. If you want to get just a small taste of pure mathematics, then you can work on the
easier problems. If you want to achieve a deeper understanding of the material, take some time to
struggle with the harder problems.

7
For instructors: This book can be used for a wide range of courses. Although the lessons can be taught
in the order presented, they do not need to be. The lessons cycle twice among eight subject areas:
logic, set theory, abstract algebra, number theory, real analysis, topology, complex analysis, and linear
algebra.
Lessons 1 through 8 give only the most basic material in each of these subjects. Therefore, an instructor
that wants to give a brief glimpse into a wide variety of topics might want to cover just the first eight
lessons in their course.
Lessons 9 through 16 cover material in each subject area that the author believes is fundamental to a
deep understanding of that particular subject.
For a first course in higher mathematics, a high-quality curriculum can be created by choosing among
the 16 lessons contained in this book.
As an example, an introductory course focusing on logic, set theory, and real analysis might cover
Lessons 1, 2, 5, 9, 10, and 13. Lessons 1 and 9 cover basic sentential logic and proof theory, Lessons 2
and 10 cover basic set theory including relations, functions, and equinumerosity, and Lessons 5 and 13
cover basic real analysis up through a rigorous treatment of limits and continuity. The first three lessons
are quite basic, while the latter three lessons are at an intermediate level. Instructors that do not like
the idea of leaving a topic and then coming back to it later can cover the lessons in the following order
without issue: 1, 9, 2, 10, 5, and 13.
As another example, a course focusing on algebraic structures might cover Lessons 2, 3, 4, 5, 10, and
11. As mentioned in the previous paragraph, Lessons 2 and 10 cover basic set theory. In addition,
Lessons 3, 4, 5, and 11 cover semigroups, monoids, groups, rings, and fields. Lesson 4, in addition to a
preliminary discussion on rings, also covers divisibility and the principle of mathematical induction.
Similarly, Lesson 5, in addition to a preliminary discussion on fields, provides a development of the
complete ordered field of real numbers. These topics can be included or omitted, as desired. Instructors
that would also like to incorporate vector spaces can include part or all of Lesson 8.
The author strongly recommends covering Lesson 2 in any introductory pure math course. This lesson
fixes some basic set theoretical notation that is used throughout the book and includes some important
exposition to help students develop strong proof writing skills as quickly as possible.
The author welcomes all feedback from instructors. Any suggestions will be considered for future
editions of the book. The author would also love to hear about the various courses that are created
using these lessons. Feel free to email Dr. Steve Warner with any feedback at
[e-mail]

8
LESSON 1 – LOGIC
STATEMENTS AND TRUTH

Statements with Words
A statement (or proposition) is a sentence that can be true or false, but not both simultaneously.
Example 1.1: “Mary is awake” is a statement because at any given time either Mary is awake or Mary
is not awake (also known as Mary being asleep), and Mary cannot be both awake and asleep at the
same time.
Example 1.2: The sentence “Wake up!” is not a statement because it cannot be true or false.
An atomic statement expresses a single idea. The statement “Mary is awake” that we discussed above
is an example of an atomic statement. Let’s look at a few more examples.
Example 1.3: The following sentences are atomic statements:
1. 17 is a prime number.
2. George Washington was the first president of the United States.
3. 5 > 6.
4. David is left-handed.
Sentences 1 and 2 above are true, and sentence 3 is false. We can’t say for certain whether sentence 4
is true or false without knowing who David is. However, it is either true or false. It follows that each of
the four sentences above are atomic statements.
We use logical connectives to form compound statements. The most commonly used logical
connectives are “and,” “or,” “if…then,” “if and only if,” and “not.”
Example 1.4: The following sentences are compound statements:
1. 17 is a prime number and 0 = 1.
2. Michael is holding a pen or water is a liquid.
3. If Joanna has a cat, then fish have lungs.
4. Albert Einstein is alive today if and only if 5+7 = 12.
5. 16 is not a perfect square.
Sentence 1 above uses the logical connective “and.” Since the statement “0 = 1” is false, it follows that
sentence 1 is false. It does not matter that the statement “17 is a prime number” is true. In fact, “T
and F” is always F.
Sentence 2 uses the logical connective “or.” Since the statement “water is a liquid” is true, it follows
that sentence 2 is true. It does not even matter whether Michael is holding a pen. In fact, “T or T” is
always true and “F or T” is always T.

9
It’s worth pausing for a moment to note that in the English language the word “or” has two possible
meanings. There is an “inclusive or” and an “exclusive or.” The “inclusive or” is true when both
statements are true, whereas the “exclusive or” is false when both statements are true. In
mathematics, by default, we always use the “inclusive or” unless we are told to do otherwise. To some
extent, this is an arbitrary choice that mathematicians have agreed upon. However, it can be argued
that it is the better choice since it is used more often and it is easier to work with. Note that we were
assuming use of the “inclusive or” in the last paragraph when we said, “In fact, “T or T” is always true.”
See Problem 4 below for more on the “exclusive or.”
Sentence 3 uses the logical connective “if…then.” The statement “fish have lungs” is false. We need to
know whether Joanna has a cat in order to figure out the truth value of sentence 3. If Joanna does have
a cat, then sentence 3 is false (“if T, then F” is always F). If Joanna does not have a cat, then sentence
3 is true (“if F, then F” is always T).
Sentence 4 uses the logical connective “if and only if.” Since the two atomic statements have different
truth values, it follows that sentence 4 is false. In fact, “F if and only if T” is always F.
Sentence 5 uses the logical connective “not.” Since the statement “16 is a perfect square” is true, it
follows that sentence 5 is false. In fact, “not T” is always F.
Notes: (1) The logical connectives “and,” “or,” “if…then,” and “if and only if,” are called binary
connectives because they join two statements (the prefix “bi” means “two”).
(2) The logical connective “not” is called a unary connective because it is applied to just a single
statement (“unary” means “acting on a single element”).
Example 1.5: The following sentences are not statements:
1. Are you happy?
2. Go away!
3. 𝑥−5 = 7
4. This sentence is false.
5. This sentence is true.
Sentence 1 above is a question and sentence 2 is a command. Sentence 3 has an unknown variable – it
can be turned into a statement by assigning a value to the variable. Sentences 4 and 5 are
self-referential (they refer to themselves). They can be neither true nor false. Sentence 4 is called the
Liar’s paradox and sentence 5 is called a vacuous affirmation.
Statements with Symbols
We will use letters such as 𝑝, 𝑞, 𝑟, and 𝑠 to denote atomic statements. We sometimes call these letters
propositional variables, and we will generally assign a truth value of T (for true) or F (for false) to each
propositional variable. Formally, we define a truth assignment of a list of propositional variables to be
a choice of T or F for each propositional variable in the list.

10