Introductory Mathematics: Algebra and Analysis

This text provides a lively introduction to pure mathematics. It begins with sets, functions and relations, proof by induction and contradiction, complex numbers, vectors and matrices, and provides a brief introduction to group theory. It moves onto analysis, providing a gentle introduction to epsilon-delta technology and finishes with continuity and functions. The book features numerous exercises of varying difficulty throughout the text.

Contenu

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.- 5.8 Cosets.- 5.9 Cyclic Groups.- 5.10 Isomorphism.- 5.11 Homomorphism.- 6. Sequences and Series.- 6.1 Denary and Decimal Sequences.- 6.2 The Real Numbers.- 6.3 Notation for Sequences.- 6.4 Limits of Sequences.- 6.5 The Completeness Axiom.- 6.6 Limits of Sequences Revisited.- 6.7 Series.- 7. Mathematical Analysis.- 7.1 Continuity.- 7.2 Limits.- 8. Creating the Real Numbers.- 8.1 Dedekind's Construction.- 8.2 Construction via Cauchy Sequences.- 8.3 A Sting in the Tail: p-adic numbers.- Further Reading.- Solutions.