This textbook is aimed at transitioning high-school students who have already developed proficiency in mathematical problem solving from numerical-answer problems to proof-based mathematics. It serves to guide students on how to write and understand mathematical proofs. It covers proof techniques that are commonly used in several areas of mathematics, especially number theory, combinatorics, and analysis. In addition to just teaching the mechanics of proofs, this book showcases key materials in these areas, thus introducing readers to interesting mathematics along with proof techniques.
Contents:
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Preliminaries:
- Sets and Set Operations
- Logic
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Number Theory:
- Proof by Contradiction
- Induction
- Prime Numbers
- The Pigeonhole Principle
- Equivalence Relations
- Congruences and Modular Arithmetic
- Modular Multiplication and Division
- Fermat's Little Theorem
- Pythagorean Triples
- Sums of Two Squares
- An Introduction to p-adic Numbers
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Combinatorics:
- Additive and Multiplicative Problems
- The Pigeonhole Principle and Matching Problems
- Double Counting
- Introduction to Binomial Coefficients
- Lattice Paths
- The Binomial Theorem
- Inclusion-Exclusion
- Recurrences
- Stirling Numbers
- The Twelvefold Way
- Counting Labeled Trees
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Analysis:
- Countable and Uncountable Sets
- Set-Theoretic Odds and Ends
- Equivalence Classes of Pairs
- Dedekind Cuts
- Sequences in ?
- A Second Construction of ?
- Limits of Sequences
- Functions and Continuity
- The Intermediate and Extreme Value Theorems
- Infinite Series
Readership: High-school students with some problem-solving background, for example, at the level of being able to qualify for the American Invitational Mathematics Exam. It can also be used as a textbook for an undergraduate introduction to proofs for honors students, and it may be of interest to mathematical hobbyists who just want to learn more mathematics rigorously.
Key Features:
- This book not only focuses on the mechanics of proofs; it differs from others in the genre by introducing a large amount of interesting mathematics along the way to pique the interest of students who are already enthusiastic about mathematics
- Highlights include determining which numbers are sums of two squares, matching problems, and two constructions of the real numbers
- This book has been used by the author to teach high-school students every year since 2018, and several hundred students have taken the classes that go along with it, so it has been successfully tested in the classroom
