The Search for Truth
In math, as in life, proof gets rid of any doubts we have that something is true. But proofs in math don’t just aim to prove one thing or one situation. They use what is called deductive reasoning to prove that something is always true in all variations for all of time. This deductive reasoning means using other proven statements to deduce new facts and proofs. As long as these original statements are indeed true, then proofs give us absolute certainty about the topic. Proofs in math might be the only time such absolute certainty is possible in life. This bold mission and staunch claim to certainty is what makes proofs so treasured in the field. In this book you’ll learn the basics of the logic, language, methods, and theories involved in making sound proofs in math.
Introductory classes in proofs are often where students first encounter pure math subjects; preceding classes usually include college-level calculus and linear algebra which relate more to science and engineering. This book aims to ease the sometimes difficult transition by providing the most thorough supplemental material possible. It lays out the topics in great detail with many examples and exercises to ensure you fully understand the unique concepts it presents.
Proofs: What’s Inside
In the first chapter, you’ll learn about the main ideas of logic that support proofs. This includes propositional logic, logical connectives, truth tables, contradictions, and more. These ideas are also useful for anyone studying philosophy and computer science. Once you have these ideas down, Chapter 1 shows you how to use them for writing actual proofs. At first, you’ll focus on proofs by example where you get a step-by-step walk through.
The second chapter looks less at examples and more at writing proofs yourself using the understanding you gained from Chapter 1. This means focusing on number theory and intuition. The chapter also covers induction, and finishes off with the Fundamental Theorem of Arithmetic. This theorem, which involves unique-prime-factorization, is arguably the most important theorem in number theory.
Chapter 3 then introduces functions, starting with sets. It includes set-builder notation, subsets, unions, intersections, and how to prove statements. Like Chapter 3, this chapter is very abstract and involves general ideas and theorems rather than actual numbers. For instance, you’ll find many examples on ideas such as Set Theory, DeMorgan’s Law, domain and codomain, and different properties of functions.
The final chapter is what really sets this book apart from many others you’ll find on the topic of proofs. In it, you’ll find an introduction to relations with meticulous detail and clarity. This includes very important equivalence relations, as well as confluent relations and the Buchberger-Winkler’s property. Many of these concepts are great for building the math foundations needed in computer science, so keep an open mind about how this topic can lift you to new heights in your academic career.