Fifth Edition, 2010
Esty and Norah C. Esty
This page updated June 30, 2012
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
We invite college and university instructors of a "proof" or
"transition to higher mathematics" course to request
- read with comprehension
- express mathematical
- reason logically
- recognize and employ
common patterns of mathematical thought
- read and write proofs
section, as a pdf file.
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."
text is designed to be used at the sophomore
before students take upper-division math courses. Capable freshman
take it. The facts of calculus are not a
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.
This text is designed for
teachers and mathematics majors.
It covers the entire
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
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
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
they have much better chances of doing the hard and clever proofs right
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"
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
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.
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.
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 n2 is even, then n is
even. [Use the contrapositive.]
Conjecture: If S is bounded, then Sc
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,
> 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
There are many conjectures that look likely (and nevertheless may be
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
Many basic properties of absolute values are derived from its
after properties of inequalities are derived. (These
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(10n)+b is divisible by 3."
There are many existence proofs.
Chapters 4 through 9 go deeper into selected topics.
Proofs in set theory, bounds, inequalities, absolute values, one-to-one
and onto, properties of f(S)
and f -1(S), number theory, and many other
areas (see the Table of Contents discussed
This text has been used at Montana
State University by Prof. Warren Esty and Prof. David Yopp, at
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, at Fitchberg State University (Massachusetts)
by Prof. Jenn Berg, and 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. Chapters
have been used by other professors. Instructors at other
are invited to request an
examination copy of the fifth edition. One advantage of
using this book is that it will cost your bookstore only a fraction of
cost of other texts (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!
second reason for using this text is that it is the best!
(In our humble opinion.)
Preview of proofs, sets, logic (including truth tables) with emphasis
on the key logical equivalences used in proofs.
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
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,
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.
Chapters 4 through 8 continue the discussion of proof by providing
practice. Each chapter is on a particular
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
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,
applied to sets [f(S) and f -1(T)], cardinality.
------------ The one-semester course at Montana State University stops
here. ----------- 266 pages to here. If you go faster than
we do, you may select from numerous additional topics in following
------------ 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:
Number Theory. Common divisors, prime numbers,
modular arithmetic, cryptography (RSA).
Group Theory. Groups. Subgroups, cosets, Lagrange's Theorem,
Isomorphism, Quotient Groups.
Topology. Open and closed sets, interior points,
accumulation (limit) points.
Calculus. Limits of sequences. Limits and
derivatives. [The basic theorems of an "advanced calculus" or "real
---------------------- 411 pages overall
Introduction to Higher Mathematics, Fifth Edition. Bright
yellow! Tape bound. 10" x 7 1/2," 411 pages.
Warren Esty (Ph.D. University of Wisconsin-Madison) is at Montana State
University in Bozeman, Montana, and Prof. Norah Esty (Ph.D., University
of California-Berkeley) is at Stonehill College outside Boston. Warren
Esty did his degree in probability theory and publishes in
probability, statistics, and math education. Norah Esty did her degree
in dynamical systems and publishes in topology.
We do not
recommend any proof text for self study. However, more than any other
text, this one gives
the reader many chances to discover erroneous reasoning and to learn to
reason well. Nevertheless, we think 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
This text covers more about how to read math than
similar texts, and provides more about all the preliminaries to proofs.
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
To request a desk copy, or merely to inquire, faculty
may write me, Warren Esty, using my address: