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cummings j proofs long form mathematics textbook 2021

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Textbook in PDF format This textbook is designed for students. Rather than the typical definition-theorem-proof-repeat style, this text includes much more commentary, motivation and explanation. The proofs are not terse, and aim for understanding over economy. Furthermore, dozens of proofs are preceded by "scratch work" or a proof sketch to give students a big-picture view and an explanation of how they would come up with it on their own.This book covers intuitive proofs, direct proofs, sets, induction, logic, the contrapositive, contradiction, functions and relations. The text aims to make the ideas visible, and contains over 200 illustrations. The writing is relaxed and conversational, and includes periodic attempts at humor.This text is also an introduction to higher mathematics. This is done in-part through the chosen examples and theorems. Furthermore, following every chapter is an introduction to an area of math. These include Ramsey theory, number theory, topology, sequences, real analysis, big data, game theory, cardinality and group theory.After every chapter are "pro-tips," which are short thoughts on things I wish I had known when I took my intro-to-proofs class. They include finer comments on the material, study tips, historical notes, comments on mathematical culture, and more. Also, after each chapter's exercises is an introduction to an unsolved problem in mathematics.In the first appendix we discuss some further proof methods, the second appendix is a collection of particularly beautiful proofs, and the third is some writing advice. Intuitive Proofs Chessboard Problems Naming Results The Pigeonhole Principle Bonus Examples Exercises Direct Proofs Working From Definitions Proofs by Cases Divisibility Greatest Common Divisors Modular Arithmetic Bonus Examples Exercises Sets Definitions Proving A B Proving A = B Set Operations Two Final Topics Bonus Examples Exercises Induction Dominoes, Ladders and Chips Examples Strong Induction Non-Examples Bonus Examples Exercises Logic Statements Truth Tables Quantifiers and Negations Proving Quantified Statements Paradoxes Bonus Examples Exercises The Contrapositive Finding the Contrapositive of a Statement Proofs Using the Contrapositive Bonus Examples Exercises Contradiction Two Warm-Up Examples Examples The Most Famous Proof in History The Pythagoreans Bonus Examples Exercises Introduction to Game Theory Functions Approaching Functions Injections, Surjections and Bijections The Composition Invertibility Bonus Examples Exercises Introduction to Cardinality Relations Equivalence Relations Abstraction and Generalization Bonus Examples Exercises Introduction to Group Theory

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Cummings J. Proofs. A Long-Form Mathematics Textbook 2021.pdf 10.8 MB

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cummings j proofs long form mathematics textbook 2021

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