eBook - ePub
Proofs 101
An Introduction to Formal Mathematics
Joseph Kirtland
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- English
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eBook - ePub
Proofs 101
An Introduction to Formal Mathematics
Joseph Kirtland
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About This Book
Proofs 101: An Introduction to Formal Mathematics serves as an introduction to proofs for mathematics majors who have completed the calculus sequence (at least Calculus I and II) and a first course in linear algebra.
The book prepares students for the proofs they will need to analyze and write the axiomatic nature of mathematics and the rigors of upper-level mathematics courses. Basic number theory, relations, functions, cardinality, and set theory will provide the material for the proofs and lay the foundation for a deeper understanding of mathematics, which students will need to carry with them throughout their future studies.
Features
- Designed to be teachable across a single semester
- Suitable as an undergraduate textbook for Introduction to Proofs or Transition to Advanced Mathematics courses
- Offers a balanced variety of easy, moderate, and difficult exercises
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CHAPTER 1
Logic
1.1 INTRODUCTION
Logic is at the foundation of everything we do in mathematics. It provides a formal language that is used to state and process results and a system of deductive reasoning used to justify or prove them. With the goal of this book in mind, the tools learned from logic are essential for crafting, writing, and analyzing proofs.
This chapter introduces a few of the basic concepts in logic. While this is not an in-depth study, it does provide all of the tools we will need in subsequent chapters. However, once you finish reading this chapter and are interested in learning more about logic, I suggest [O'L16] and [GU89].
1.2 STATEMENTS AND LOGICAL CONNECTIVES
Statements are the basic elements of logic.
Definition 1.2.1. A statement is a declarative sentence that is either true or false, but not both.
To simplify the calculus of logic, the letters are used to denote statements. The following sentences are all statements.
The Empire State Building is in New York City. | |
The digit of is 3. |
The statement is false, is true, and , while we may not know the digit of , is definitely either true or false.
Sentences such as, Are you tired, , and The Empire State Building is tall, are not statements. The first is not a statement as it ...