Pure Mathematics for Beginners A Rigorous Introduction to Logic, Set Theory, Abstract Algebra, Number Theory, Real Analysis, Topology, Complex Analysis, and Linear Algebra by Steve Warner (z-lib.org)

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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

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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
© 2018, All Rights Reserved

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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  enough? 56
Completeness 58
Problem Set 5 62
Lesson 6 – Topology: The Topology of  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  85
Problem Set 7 90

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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