This book is designed for the transition course between calculus and differential equations and the upper division mathematics courses with an emphasis on proof and abstraction. The book has been used by the author and several other faculty at Southern Connecticut State University. There are nine chapters and more than enough material for a semester course. Student reviews are favorable.
It is written in an informal, conversational style with a large number of interesting examples and exercises, so that a student learns to write proofs while working on engaging problems. The lecture videos can be used for a “flipped course” or for students’ additional review.
- Title
- A Gentle Introduction to the Art of Mathematics, version 3.2
- Author(s)
- Joe Fields
- Published
- 2023-08-22
- Edition
- 1
- Format
- eBook (pdf, epub, mobi)
- Pages
- 434
- Language
- English
- ISBN-10
- B0CG87V5F9
- ISBN-13
- 9798858311546
- License
- GNU FDL
- Book Homepage
- Free eBook, Errata, Code, Solutions, etc.
Introduction and notation Basic sets Definitions: Prime numbers More scary notation Definitions of elementary number theory Even and odd Decimal and base-n notation Divisibility Floor and ceiling Div and mod Binomial coefficients Some algorithms Rational and irrational numbers Relations Logic and quantifiers Predicates and Logical Connectives Implication Logical equivalences Two-column proofs Quantified statements Deductive reasoning and Argument forms Validity of arguments and common errors Proof techniques I Direct proofs of universal statements More direct proofs Contradiction and contraposition Disproofs By cases and By exhaustion Existential statements Sets Basic notions of set theory Containment Set operations Venn diagrams Russell's Paradox Proof techniques II — Induction The principle of mathematical induction Formulas for sums and products Other proofs using PMI The strong form of mathematical induction Relations and functions Relations Properties of relations Equivalence relations Ordering relations Functions Special functions Proof techniques III — Combinatorics Parity and Counting arguments The pigeonhole principle The algebra of combinations Cardinality Equivalent sets Examples of set equivalence Cantor's theorem Dominance CH and GCH Proof techniques IV — Magic Morley's miracle Five steps into the void Monge's circle theorem References Index
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