| Preface 1 Sets, Functions and Relations 1.1 Sets 1.2 Subsets 1.3 Well-known Sets 1.4 Rationals, Reals and Pictures 1.5 Set Operations 1.6 Sets of Sets 1.7 Paradox 1.8 Set-theoretic Constructions 1.9 Notation 1.10 Venn Diagrams 1.11 Quantifiers and Negation 1.12 Informal Description of Maps 1.13 Injective, Surjective and Bijective Maps 1.14 Composition of Maps 1.15 Graphs and Respectability Reclaimed 1.16 Characterizing Bijections 1.17 Sets of Maps 1.18 Relations 1.19 Intervals 2 Proof 2.1 Induction 2.2 Complete Induction 2.3 Counter-examples and Contradictions 2.4 Method of Descent 2.5 Style 2.6 Implication 2.7 Double Implication 2.8 The Master Plan 3 Complex Numbers and Related Functions 3.1 Motivation 3.2 Creating the Complex Numbers 3.3 A Geometric Interpretation 3.4 Sine, Cosine and Polar Form 3.5 e 3.6 Hyperbolic Sine and Hyperbolic Cosine 3.7 Integration Tricks 3.8 Extracting Roots and Raising to Powers 3.9 Logarithm 3.10 Power Series 4 Vectors and Matrices 4.1 Row Vectors 4.2 Higher Dimensions 4.3 Vector Laws 4.4 Lengths and Angles 4.5 Position Vectors 4.6 Matrix Operations 4.7 Laws of Matrix Algebra 4.8 Identity Matrices and Inverses 4.9 Determinants 4.10 Geometry of Determinants 4.11 Linear Independence 4.12 Vector Spaces 4.13 Transposition 5 Group Theory 5.1 Permutations 5.2 Inverse Permutations 5.3 The Algebra of Permutations 5.4 The Order of a Permutation 5.5 Permutation Groups 5.6 Abstract Groups 5.7 Subgroups …… 6 Sequences and Series 7 Mathematical Analysis 8 Creating the Real Numbers Further Reading Solutions Index |
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