Seventh Edition

by Warren W. Esty and Norah C. Esty

This page updated Oct. 27, 2023

This page updated Oct. 27, 2023

This is a text for a course that introduces math and math-education majors to the concepts, reasoning patterns, and language skills that are fundamental to higher mathematics—especially proofs.

After many years and seven paper editions, we have decided to make the text available **for free **as a pdf. Take a look at these three things. If you are still interested after reading them, contact us for the entire pdf. Contact info is at the bottom of this page.

1) Here is the Table of Contents in pdf, including the Preface and "To the Instructor." The sections are discussed below.

2) Section 1.1, a preview section, as a pdf file.

3) Another sample section, Section 2.2 on "Existence Statements and Negation."

**Level**. This text is designed to be used at the sophomore level before students take upper-division math courses. Capable freshman could take it. Calculus is not a prerequisite. The text does not assume that the students are already strong in logic or the underlying language of mathematics; rather, it aims to make them so.

**Approach**. This text is designed for mathematics majors and future mathematics teachers. It covers the entire language of mathematics including the uses of variables, the conventions of the language, logic, and all the other language features involved in proofs. Many students can **do** mathematical procedures well, but many cannot yet **read or write** the language well. Our approach is to include many reading and writing lessons and activities before expecting creative proofs from students.

**Supplements.** If you are an instructor using our text, we will share with you many old exams and quizzes, handouts, and solutions manuals (an Instructor's Solutions Manual for the instructor only, and pdf solution files by chapter to offer—or not—to students), as well as an Instructor's Manual. There's a website showing the assignments and pace I used, as well as the advice I gave, when I last taught the class.

**Cost**. **Free** as a **pdf**. We have a few paper copies which are reserved for instructors. Anyone, instructors and students, may have the pdf printed and bound.

**What Sort of Proofs? **Mostly basic proofs—many of them. We want students to know how to organize and get the straightforward proofs right. If they can, they have much better chances of doing the hard and clever proofs. Proofs are about such topics as sets, bounds, one-to-one and onto functions, limits, and other topics listed here.

Here is a link to short page on why faculty would prefer to use this text for a "proof" course.

A university professor commented, "I definitely do not like the other two [introduction to Proof] books I tried. ... The main reason I would use it [The Esty text] again was because it was easy to make class fun when I used your book. We did a lot of fast moving things in class with the quick response items, there were portions of the book that were great for some group work, and the students learned (and enjoyed) the false proofs." [Continued here.]

Here is a link to a page on Unexpected Things We Do that You will Like and Appreciate.

**Evidence of our attitude**:

**Homework**. Homework problems are categorized as A (basic), B (the current level of the course), or C (not required and more advanced, but using the ideas of the section). Many A problems are marked with this symbol:** ☺**. These problems are simple and very short. Nevertheless, they are conceptually important. Going over problems aloud in class helps the class learn these essential basics, and helps you, the instructor, learn how uncomfortable or comfortable the students are. They can be used anytime during class to make sure students got it.

Proofs early in the text emphasize logical organization and using definitions properly:

- Theorem: If S ∩ T = S, then S ⊂ T. [Why might a proof begin, "Let x ∈ S"? Translation and logic suggest that would be a good place to begin.]
- Theorem (half of The Zero Product Rule): If xy = 0, then x = 0 or y = 0. [Why can we prove "If xy = 0 and x ≠ 0, then y = 0" instead?]
- Theorem: If n
^{2}is even, then n is even. [Use the contrapositive.] - Conjecture: If S is bounded, then S
^{c}is not bounded. [You need to translate "not bounded" into more useful terminology.] - Conjecture: If f is increasing and g(x) = f(2x), then g is increasing. [You need to translate "increasing".]
- Conjecture: If x and y are both irrational, then xy is irrational. [Give a counterexample.]
- Conjecture: If x is rational and y is irrational, then x+y is irrational. [Use a version of the contrapositive to prove it.]

**We include many conjectures** that look likely (but may be false) and ask students to resolve them:

- Conjecture: If x < y, then x
^{2 }< y^{2}. - Conjecture: |x| < |x+1|.
- Conjecture: If |a| < |b| then |a + c| < |b + c|.
- Conjecture: If f(x) ∈ f(S), then x ∈ S.

We discuss truth tables, but primarily to illuminate the meanings of the connectives (especially "if..., then...") and to prove a few important logical equivalences that are frequently used to organize proofs.

The syntax of nested quantifiers is discussed at length:

- Conjecture: Let S = (0, 1). For all x ∈ S there exists y ∈ S such that y > x.
- Conjecture: Let S = (0, 1). There exists y ∈ S such that for all x ∈ S, y > x. [Why do these two, with the same words, have different meanings and truth values?]
- Conjecture: Let S = [0, 1]. If x ∈ S there exists y ∈ S such that y > x.

Inequalities and absolute values are used a great deal in Advanced Calculus/Real Analysis courses, so we emphasize them. They are good subjects for low-level basic proofs and learning to play by the rules.

There are many basic induction proofs (and a few harder ones). "If *a*+*b* is divisible by 3, then *a*(10^{n})+*b* is divisible by 3."

There are many existence proofs.

**Courses**: This text has been used at

- Montana State University
- University of Idaho, Moscow
- Marshall University
- Texas State University, San Marcos
- Stonehill College (Boston)
- Fitchberg State University (Massachusetts)
- Case Western University (Cleveland, Ohio)
- Boise State University (Idaho)
- Augustana University (Sioux Falls, SD)
- University of Minnesota, Duluth

**Contents:**

**Chapter 1**: Preview of proofs (Here is Section 1.1, "Preview of Proof"), sets, logic (including truth tables) with emphasis on the key logical equivalences used in proofs.

**Chapter 2**: Uses of variables, generalizations, existence statements (Here is Section 2.2, "Existence Statements and Negation"), negations, how to read theorems and definitions, how the forms of statements can be rearranged, and how to work with recently-defined terms.

**Chapter 3**: Proofs, in general. Representative-case proofs, existence proofs, proofs by contrapositive and contradiction, and proofs by induction. Proofs of basic facts about inequalities and absolute values-- areas which are just tricky enough that mistakes occur frequently. The absolute value section has, interspersed with theorems, numerous "conjectures," some true and some false, which help students become critical thinkers. The final section in the chapter, "Bad Proofs," requires students to judge arguments and recognize some of the most common types of errors.

**Chapters 1 through 3 provide a complete discussion of the language of mathematics and the theory of proofs.**

**Chapters 4 through 8 continue the discussion of proof by providing practice**.

The instructor need not do these chapters in order.

***** Part II:

**Chapter 4**: Set theory, bounds, supremum.

**Chapter 5**: Functions, one-to-one, onto, bijection, functions applied to sets [*f*(*S*) and *f*^{ -1}(*T*)], cardinality.

**The one-semester course at Montana State University ends here, **however, you may select from numerous additional topics in following chapters.

**Chapter 6**: Number Theory. Common divisors, prime numbers, modular arithmetic, cryptography (RSA).

**Chapter 7**: Group Theory. Groups. Subgroups, cosets, Lagrange's Theorem, Isomorphism, Quotient Groups.

**Chapter 8**: Topology. Open and closed sets, interior points, accumulation (limit) points.

**Chapter 9**: Calculus. Limits of sequences. Limits and derivatives. [The basic theorems of an advanced calculus or real analysis course.]

**The Authors**. Prof. Warren Esty (Ph.D., University of Wisconsin-Madison) is a Professor Emeritus at Montana State University in Bozeman, Montana, and Prof. Norah Esty (Ph.D., University of California-Berkeley) retired from Stonehill College outside Boston. Warren Esty did his degree in probability theory and published in probability, statistics, and math education (see here for more). Norah Esty did her degree in dynamical systems and published in topology.

**Requesting a Copy**. Please read (at least) Section 1.1 before contacting me. Then, to request a pdf copy, or merely to inquire, write me, Warren Esty, at: