\documentclass{article} \usepackage{latexsym} \author{Geoff Smith} \title{Introductory Mathematics: Algebra and Analysis} \author{G.~C.~Smith} \begin{document} \maketitle \noindent {\bf Preface} \hfill {v}\\ \\ {\bf 1. Sets, Functions and Relations} \hfill {1}\\ {1.1} Sets\hfill{1}\\ {1.2} Subsets\hfill{2}\\ {1.3} Well-known Sets\hfill{3}\\ {1.4} Rationals, Reals and Pictures\hfill{6}\\ {1.5} Set Operations\hfill{8}\\ {1.6} Sets of Sets\hfill{11}\\ {1.7} Paradox\hfill{14}\\ {1.8} Set-theoretic Constructions\hfill{15}\\ {1.9} Notation\hfill{16}\\ {1.10} Venn Diagrams\hfill{17}\\ {1.11} Quantifiers and Negation\hfill{19}\\ {1.12} Informal Description of Maps\hfill{21}\\ {1.13} Injective, Surjective and Bijective Maps\hfill{22}\\ {1.14} Composition of Maps\hfill{23}\\ {1.15} Graphs and Respectability Reclaimed\hfill{29}\\ {1.16} Characterizing Bijections\hfill{30}\\ {1.17} Sets of Maps\hfill{31}\\ {1.18} Relations\hfill{31}\\ {1.19} Intervals\hfill{37}\\ \\ {\bf 2. Proof}\hfill{39}\\ {2.1} Induction\hfill{39}\\ {2.2} Complete Induction\hfill{43}\\ {2.3} Counter-examples and Contradictions\hfill{47}\\ {2.4} Method of Descent\hfill{50}\\ {2.5} Style\hfill{53}\\ {2.6} Implication\hfill{54}\\ {2.7} Double Implication\hfill{54}\\ {2.8} The Master Plan\hfill{56}\\ \eject \noindent {\bf 3. Complex Numbers and Related Functions}\hfill{57}\\ {3.1} Motivation\hfill{57}\\ {3.2} Creating the Complex Numbers\hfill{62}\\ {3.3} A Geometric Interpretation\hfill{70}\\ {3.4} Sine, Cosine and Polar Form\hfill{76}\\ {3.5} $e$\hfill{80}\\ {3.6} Hyperbolic Sine and Hyperbolic Cosine\hfill{85}\\ {3.7} Integration Tricks\hfill{88}\\ {3.8} Extracting Roots and Raising to Powers\hfill{89}\\ {3.9} Logarithm\hfill{90}\\ {3.10} Power Series\hfill{92}\\ \\ \noindent {\bf 4. Vectors and Matrices}\hfill{95}\\ {4.1} Row Vectors\hfill{95}\\ {4.2} Higher Dimensions\hfill{97}\\ {4.3} Vector Laws\hfill{98}\\ {4.4} Lengths and Angles\hfill{98}\\ {4.5} Position Vectors\hfill{103}\\ {4.6} Matrix Operations\hfill{104}\\ {4.7} Laws of Matrix Algebra\hfill{106}\\ {4.8} Identity Matrices and Inverses\hfill{108}\\ {4.9} Determinants\hfill{110}\\ {4.10} Geometry of Determinants\hfill{119}\\ {4.11} Linear Independence\hfill{120}\\ {4.12} Vector Spaces\hfill{121}\\ {4.13} Transposition\hfill{123}\\ \\ \noindent {\bf 5. Group Theory}\hfill{125}\\ {5.1} Permutations\hfill{125}\\ {5.2} Inverse Permutations\hfill{130}\\ {5.3} The Algebra of Permutations\hfill{131}\\ {5.4} The Order of a Permutation\hfill{133}\\ {5.5} Permutation Groups\hfill{135}\\ {5.6} Abstract Groups\hfill{136}\\ {5.7} Subgroups\hfill{142}\\ {5.8} Cosets\hfill{143}\\ {5.9} Cyclic Groups\hfill{146}\\ {5.10} Isomorphism\hfill{148}\\ {5.11} Homomorphism\hfill{151}\\ \eject \noindent {\bf 6. Sequences and Series}\hfill{153}\\ {6.1} Denary and Decimal Sequences\hfill{153}\\ {6.2} The Real Numbers\hfill{156}\\ {6.3} Notation for Sequences\hfill{158}\\ {6.4} Limits of Sequences\hfill{160}\\ {6.5} The Completeness Axiom\hfill{169}\\ {6.6} Limits of Sequences Revisited\hfill{170}\\ {6.7} Series\hfill{174}\\ \\ \noindent {7. Mathematical Analysis}\hfill{181}\\ {7.1} Continuity\hfill{181}\\ {7.2} Limits\hfill{194}\\ \\ \noindent {\bf 8. Creating the Real Numbers}\hfill{197}\\ {8.1} Dedekind's Construction\hfill{197}\\ {8.2} Construction via Cauchy Sequences\hfill{199}\\ {8.3} A Sting in the Tail: $p$-adic numbers\hfill{202}\\ \\ \noindent {\bf Further Reading}\hfill{204}\\ \\ \noindent {\bf Solutions}\hfill{205}\\ \\ \noindent {\bf Index}\hfill{213} \end{document}