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