CHAPTER 3
Methods of Proofs
1. Logical Arguments and Formal Proofs
1.1. Basic Terminology.
•
An
axiom
is a statement that is given to be true.
•
A
rule of inference
is a logical rule that is used to deduce one statement
from others.
•
A
theorem
is a proposition that can be proved using definitions, axioms,
other theorems, and rules of inference.
Discussion
In most of the mathematics classes that are prerequisites to this course, such
as calculus, the main emphasis is on using facts and theorems to solve problems.
Theorems were often stated, and you were probably shown a few proofs. But it is
very possible you have never been asked to prove a theorem on your own.
In this
module we introduce the basic structures involved in a mathematical proof. One of
our main objectives from here on out is to have you develop skills in recognizing a
valid argument and in constructing valid mathematical proofs.
When you are first shown a proof that seemed rather complex you may think to
yourself “How on earth did someone figure out how to go about it that way?” As we
will see in this chapter and the next, a proof must follow certain
rules of inference
,
and there are certain strategies and methods of proof that are best to use for proving
certain types of assertions.
It is impossible, however, to give an exhaustive list of
strategies that will cover all possible situations, and this is what makes mathematics
so interesting. Indeed, there are conjectures that mathematicians have spent much
of their professional lives trying to prove (or disprove) with little or no success.
1.2. More Terminology.
•
A
lemma
is a “pretheorem” or a result which is needed to prove a theorem.
•
A
corollary
is a “posttheorem” or a result which follows from a theorem
(or lemma or another corollary).
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1. LOGICAL ARGUMENTS AND FORMAL PROOFS
57
Discussion
The terms “lemma” and “corollary” are just names given to theorems that play
particular roles in a theory.
Most people tend to think of a theorem as the main
result, a lemma a smaller result needed to get to the main result, and a corollary
as a theorem which follows relatively easily from the main theorem, perhaps as a
special case.
For example, suppose we have proved the Theorem: “If the product
of two integers
m
and
n
is even, then either
m
is even or
n
is even.” Then we have
the Corollary: “If
n
is an integer and
n
2
is even, then
n
is even.” Notice that the
Corollary follows from the Theorem by applying the Theorem to the special case in
which
m
=
n
. There are no firm rules for the use of this terminology; in practice,
what one person may call a lemma another may call a theorem.
Any mathematical theory
must
begin with a collection of undefined terms and
axioms that give the properties the undefined terms are assumed to satisfy. This may
seem rather arbitrary and capricious, but any mathematical theory you will likely
encounter in a serious setting is based on concrete ideas that have been developed
and refined to fit into this setting. To justify this necessity, see what happens if you
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 Fall '08
 PenelopeKirby

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