Preface to the Student
Welcome to the study of mathematical reasoning. The authors know that many students
approach this material with some apprehension and uncertainty. Some students say that “this
isn’t like other mathematics courses,” or expect that “I’ll never use this later.” These feelings are
natural as you move from calculation oriented courses where the goal may be to solve certain
equations, to more advanced courses where the goal may be to establish whether a mathematical
structure has certain properties. This textbook is written to help ease the transition between these
courses.
Why write proofs?
Mathematicians can be distinguished from other scientists by the kind of reasoning that
we do. Natural and social scientists usually approach a problem by making observations of
particular cases or phenomena and seeking a statement (a theory) that describes or explains the
observations – an approach called inductive reasoning. The theory is then either confirmed or
rejected by making further observations. Mathematicians, too, use inductive reasoning as they
attempt to describe patterns and relationships among quantities and structures. However, the
characteristic thinking of mathematics is deductive reasoning, in which one uses logic to
conclude that the statement of the theory is true based on other statements accepted as true. The
theory is tested by giving a proof, which is just a complete explanation of why the theory is true.
Mathematicians see the ability to give a proof as the ultimate test of a theory. As long as the
statements from which the theory is derived are assumed or proven to be true, there is no doubt
about the theory – it must be true.
Why not just test a theory by trying enough examples until we’re convinced that it
works? One answer is that there are many mathematical statements that are true for lots of
examples but not always true. For example, by simply studying examples, one might conclude
that the statement “ x 2  3 x  43 is a prime number” is true for all positive integers x. One could
be influenced to think so because x 2  3 x  43 is prime for the first forty two integers, 1, 2, 3,
….42 and many integers 44, 45, … thereafter. The smallest integer for which the statement is not
true is x = 43, because 1763  432  3(43)  43  43(43  3  1)  (41)(43) .
Why not just rely on proofs that someone else has done? One answer follows from the
statement above that deductive reasoning characterizes the way mathematicians think. In other
sciences, a new observation may force a complete rethinking of what was thought to be true; in
mathematics what we know to be true (by proof) is true forever unless there was a flaw in the
reasoning. By learning the techniques of reasoning and proof you are learning the tools of the
trade.
The first goal of this text is to examine standard proof techniques, especially concentrating
on how to get started on a proof, and how to construct correct proofs using those techniques. You
will discover how the logical form of a complicated statement can serve as a guide to the
structure of a proof. As you study more advanced courses, it will become apparent that the
material in this book is indeed fundamental and the knowledge gained will help you succeed in
those courses. Moreover, many of the techniques of reasoning and proof that may seem so
1
difficult at first will become completely natural with practice. In fact, the reasoning that you will
study is the essence of advanced mathematics and the ability to reason abstractly is a primary
reason why applicants trained in mathematics are valuable to employers.
“What am I allowed to assume for this proof?”
You may be given specific instructions for some proof-writing exercises, but generally
the idea is that you can use what is known to students in the relevant course. That is, when we
prove something about intersecting lines we might use facts about the slope of a line, but we
probably wouldn’t use properties of derivatives. This really isn’t much of a problem, except for
our very first proof examples, which deal with elementary concepts such as even and odd
integers (because they provide meaningful examples and a familiar context in which to study
logic and reasoning). For these proofs we are allowed to use properties of integers and natural
numbers we already know about, EXCEPT what we already know about evenness and oddness.
Definitions in mathematics are not like definitions in ordinary English, which are based
on how words are typically used. For example, in ordinary English, the word “cool” came to
mean something good or popular when many people used it that way, not because it has to have
that meaning. If people stop using the word that way, this meaning of the word will change. This
is not the case with mathematical definitions, which are stipulated. When we say a number is
odd, we don’t mean that it’s strange or unusual. Our definition tells you exactly what odd
means. You may form a concept or a mental image that you may use to help understand (such as
“ends in 1,3,5,7, or 9”), but the mental image you form is not what has been defined.
Definitions are stated with the “if and only if” connective because they describe exactly –
no more, no less – the condition(s) to meet the definition. Some of the definitions from number
theory that will be used in our first examples of proof-writing are:
An integer x is even if and only if there is a natural number k such that x = 2k.
An integer x is odd if and only if there is a natural number j such that x = 2j + 1.
For natural numbers a and b we say a divides b if and only if there is a natural number k such
that b = ak.
A natural number p is prime if and only if p is greater than 1 and the only natural numbers
that divide p are 1 and p.
The real number x is rational if and only if there exist integers p and q, q ≠ 0, such that x =
p/q.
You will use definitions in two ways. First, if you are told that x is odd, then you know
that for some natural number k, x = 2k + 1. Secondly, if you need to prove that x is odd, then you
must show that it can be expressed as x = 2k + 1, for some natural number k. This discussion may
sound obvious or irrelevant to you now, but you’ll find it extremely useful in writing proofs if
you keep it in mind.
An Exercise for the Preface NOTE: STILL UNDER DISCUSSION WHETHER WE
WILL USE THIS HERE.
2
As noted above, many concepts have multiple, yet equivalent ways to describe them.
Some equivalents can be helpful for understanding a concept but are cumbersome for showing
that a given object fits the definition. We could have defined integer x is even if and only if x
ends in 0, 2, 4, 6, or 8 as its last digit. This equivalent way to define “even” actually works well
if we are given a specific number and asked whether it is even. However, if we were asked
“Prove that if x and y are integer solutions to 2x + y = 10, then y is even.”, we must cope with a
large number of separate cases if we had to show y ends in 0, 2, 4, 6, or 8. It would be simpler
proof to rewrite the equation as y = 10 – 2x = 2(5-x) and note that y now has the form 2k where k
is the integer 5-x.
The pairs of definitions that follow each define the concept in italics. Which definition
will be more useful when it comes time to prove that an object has the italicized property?
Reciprocal
(a) Two nonzero rational numbers r and s are reciprocals if and only if one is obtained
from the other by inverting the numerator and denominator.
(b) Two nonzero rational numbers r and s are reciprocals if and only if rs = 1.
Increasing function
(a) Let f be a function with domain and range real numbers. The function f is increasing if
and only if the graph slopes upward from left to right.
(b) Let f be a function with domain and range real numbers. The function f is increasing if
and only if f(x1) < f(x2) whenever x1 < x2.
Parallel Lines
(a) Two lines in a plane are parallel if and only if they are both vertical lines or have the
same slope.
(b) Two lines in a plane are parallel if and only if they never cross.
Solution to an equation
(a) A real number r is a solution to the equation E(x) = 0 if and only if the graph of y =
E(x) intersects the x-axis at the point r.
(b) A real number r is a solution to the equation E(x) = 0 if and only E(r) = 0.
Answers to the Exercise in the Preface to the Student
Reciprocals (b)
Increasing function (b)
Parallel Lines
(a)
Solution to an equation
(b)
While the other definitions provide good insight, for these concepts we choose the definition that
involves an equation or inequality. For example, definition (a) for reciprocals works well if you
want to say why 2/5 and 5/2 are reciprocals. However, proofs will talk about numbers in general,
not just specific numbers. Using definition (b), if we know two numbers are reciprocals then we
have an equation that relates them; this helps because we know a lot about operating with
3
equations. Secondly, to prove numbers are reciprocals, we simply show an equation is true,
rather than having to perform some manipulation on them.
4