Introductory Mathematics: Algebra and Analysis

G. C. Smith

February 5, 1998

Preface v

1. Sets, Functions and Relations 1 1.1 Sets 1 1.2 Subsets 2 1.3 Well-known Sets 3 1.4 Rationals, Reals and Pictures 6 1.5 Set Operations 8 1.6 Sets of Sets 11 1.7 Paradox 14 1.8 Set-theoretic Constructions 15 1.9 Notation 16 1.10 Venn Diagrams 17 1.11 Quantifiers and Negation 19 1.12 Informal Description of Maps 21 1.13 Injective, Surjective and Bijective Maps 22 1.14 Composition of Maps 23 1.15 Graphs and Respectability Reclaimed 29 1.16 Characterizing Bijections 30 1.17 Sets of Maps 31 1.18 Relations 31 1.19 Intervals 37

2. Proof 39 2.1 Induction 39 2.2 Complete Induction 43 2.3 Counter-examples and Contradictions 47 2.4 Method of Descent 50 2.5 Style 53 2.6 Implication 54 2.7 Double Implication 54 2.8 The Master Plan 56 3. Complex Numbers and Related Functions 57 3.1 Motivation 57 3.2 Creating the Complex Numbers 62 3.3 A Geometric Interpretation 70 3.4 Sine, Cosine and Polar Form 76 3.5 e 80 3.6 Hyperbolic Sine and Hyperbolic Cosine 85 3.7 Integration Tricks 88 3.8 Extracting Roots and Raising to Powers 89 3.9 Logarithm 90 3.10 Power Series 92

4. Vectors and Matrices 95 4.1 Row Vectors 95 4.2 Higher Dimensions 97 4.3 Vector Laws 98 4.4 Lengths and Angles 98 4.5 Position Vectors 103 4.6 Matrix Operations 104 4.7 Laws of Matrix Algebra 106 4.8 Identity Matrices and Inverses 108 4.9 Determinants 110 4.10 Geometry of Determinants 119 4.11 Linear Independence 120 4.12 Vector Spaces 121 4.13 Transposition 123

5. Group Theory 125 5.1 Permutations 125 5.2 Inverse Permutations 130 5.3 The Algebra of Permutations 131 5.4 The Order of a Permutation 133 5.5 Permutation Groups 135 5.6 Abstract Groups 136 5.7 Subgroups 142 5.8 Cosets 143 5.9 Cyclic Groups 146 5.10 Isomorphism 148 5.11 Homomorphism 151 6. Sequences and Series 153 6.1 Denary and Decimal Sequences 153 6.2 The Real Numbers 156 6.3 Notation for Sequences 158 6.4 Limits of Sequences 160 6.5 The Completeness Axiom 169 6.6 Limits of Sequences Revisited 170 6.7 Series 174

7. Mathematical Analysis 181 7.1 Continuity 181 7.2 Limits 194

8. Creating the Real Numbers 197 8.1 Dedekind’s Construction 197 8.2 Construction via Cauchy Sequences 199 8.3 A Sting in the Tail: p-adic numbers 202

Further Reading 204

Solutions 205

Index 213