Seventh Edition

by Warren W. Esty and Norah C. Esty

This page updated Nov.1, 2020

This page updated Nov.1, 2020

This is a text for a course that introduces math majors and math-education majors to the basic concepts, reasoning patterns, and language skills that are fundamental to higher mathematics. The skills include the ability to

- read with comprehension
- express mathematical thoughts clearly
- reason logically
- recognize and employ common patterns of mathematical thought
- read and write proofs

We invite college and university instructors of a "proof" or "transition to higher mathematics" course to request an examination copy (Contact info is at the bottom of this page).

Contents listed. (They are discussed below.) Section 1.1, a preview section, as a pdf file. The Table of Contents in pdf, including the Preface and "To the Instructor." Here is another sample section, Section 2.2 on "Existence Statements and Negation."

**Supplements**: If you adopt it, we will share with you a dropbox folder with many old exams and quizzes, handouts, and solutions manuals (an Instructor's Solutions Manual for the instructor only, and pdf solution files by chapter free to send out or post--or not--for students), and an Instructor's Manual. I communicated with class using a website and you are welcome to see (and perhaps duplicate) the assignments and pace I used, as well as the advice I gave, that semester.

**Level**. This text is designed to be used at the sophomore level before students take upper-division math courses. Capable freshman could take it. The facts of calculus are 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. We believe that interested students can become good at reading and writing mathematics, including proofs, if they are taught logical thinking and the notations and patterns of mathematical writing. This text, unlike other transitional mathematics textbooks, provides this necessary background, in addition to teaching methods of proof in the context of several mathematical fields.

**Approach**. This text is designed for future mathematics teachers and mathematics majors. 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. Most students who like math enough to take a course like this 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.

**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 right too.

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.

Truth table proofs of important (and only important) logical equivalences.

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

Examples such as these are reorganized using definitions and logical equivalences. There are many more variants of each type.

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.]

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.

There are many conjectures that look likely (and nevertheless may be false)

Conjecture: |x| < |x+1|.

Conjecture: If |a| < |b| then |a + c| < |b + c|

Conjecture: If f(x) ∈ f(S), then x ∈ S.

Inequalities and absolute values are used a great deal in Advanced Calculus/Real Analysis courses, so we emphasize them.

Many basic properties of > and < are derived from the trichotomy law, but not at the beginning. Section 1.1 gives some idea of what we do before students are ready for such proofs.

Many basic properties of absolute values are derived from its definition after properties of inequalities are derived. (These properties are used in calculus proofs.)

Epsilon-delta proofs are introduced in the early chapters (and there are many more, and more-sophisticated ones, in the later chapters).

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 by Prof. Warren Esty, by Prof. David Yopp at University of Idaho, Moscow, at Marshall University by Prof. Judy Silver and Prof. Alan Horwitz, at Texas State San Marcos by Prof. David Snyder, at Stonehill College outside Boston by Prof. Norah Esty and Prof. Heiko Todt, at Fitchberg State University (Massachusetts) by Prof. Jenn Berg, at Case Western University (Cleveland, Ohio) by Prof. Elizabeth Meckes and Prof. Marc Meckes, at Boise State University (Idaho) by Prof. Zach Teitler and Prof. Andres Caicedo, and at the Universityof Minnesota--Duluth by Prof. Carmen Latterell. Chapters have been used by other professors. Instructors at other schools are invited to request an examination copy. One advantage of using this book is that it will cost your bookstore only a fraction of the cost of other texts (As I write, Fletcher and Patty, a slim text for the same market, lists at $170 and is discounted on Amazon to $120!). Our text is inexpensive because it is self-published and we can provide it without the usual whopping markup. Your students will appreciate that! A second reason for using this text is that it is the best! (In our humble opinion.)

**Contents**. **Chapter 1**: Preview of proofs, sets, logic (including truth tables) with emphasis on the key logical equivalences used in proofs.

**Chapter 2**: Uses of variables, generalizations, existence statements, 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.** For a sample of what it discussed in Chapters 1-3, here is Section 1.1, a preview section that summarizes many key ideas. Here is another sample section, Section 2.2 on "Existence Statements and Negation."

**Chapters 4 through 8 continue the discussion of proof by providing practice**. Each chapter is on a particular topic--set theory, functions, number theory, group theory, topology, and calculus.

The instructor need not do these chapters in order. Pick your favorites. No chapter goes very deeply into the subject, but each goes deep enough to yield interesting results and many good examples of proofs and conjectures. The use of many conjectures is a characteristic of this text. The students have to do a lot of thinking!

***** 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**. ----------- 275 pages to here. If you go faster than we do, you may select from numerous additional topics in following chapters.

------------ Montana State University has had a **second semester** for future teachers who do not have time in their curriculum for an entire course on each of these topics. When we cover number theory, abstract algebra, etc., it resumes here:

**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.]

---------------------- 426 pages overall, but MSU used only the first 275 in one three-credit semester-long course.

*Proof: Introduction to Higher Mathematics*, Seventh Edition. Bright yellow! Perfect bound. 10" x 7 1/2," 426 pages.

**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 publishes in probability, statistics, and math education (see here for more). Norah Esty did her degree in dynamical systems and published in topology.

**Self-Study? **We do not sell our text to individual students. We do not recommend any proof text for self study. It takes an instructor to judge the work of students, for it is impossible for students to recognize, on their own, when their line of thought is not logical. That's why we study logic!

This text covers more about how to read math than other similar texts, and provides more about all the preliminaries to proofs. More than any other text, this one gives the student many chances to discover erroneous reasoning and to learn to reason well. But this text has many "conjectures," which are plausible statements that may or may not be true. These conjectures are used to force students to think critically and to not make false assumptions. Experience shows that even though the conjectures are simple and very closely related to current work in the text, untrained students often do not know which of these are true and which are false. Or, they often provide illogical reasons for their conclusions. Students need an instructor who can respond to their work and correct erroneous thinking.

Therefore, we do not sell copies to students. However, Warren Esty has written a lower-level book, *The Language of Mathematics*, that covers all the logic, but so much more thoroughly and with such an extensive solutions manual that we do sell it to individuals. They get a lot out of it. However, it does not cover any of the advanced topics covered in *Proof*. You can learn about *The Language of Mathematics* here.

**Requesting a Copy**. To request a desk copy, or merely to inquire, **faculty** may write me, Warren Esty, using my address: