Why Read This Book? - Index of

40 Chapter 1 Language and Mathematics
1.5 How We Write Proofs
A theorem is a statement of the form U → V . To “prove” a theorem in the most
abstract sense is to show that U → V is a tautology, for then we know that U is at
least as strong as V . Thus, as we say, V follows from U, meaning that any context
in which U is true will guarantee that V is also true.
Only the most theoretical mathematicians approach theorems as tautologies,
and even then such mathematicians are probably more interested either in the
philosophy of mathematical reasoning or in formal methods as it relates to fields
such as computer science. In this text, we do hope that some mathematical philosophy
rubs off on you. But we are primarily interested in proofs as almost all
mathematicians write them. Thus we focus on what it means to write a proof in
mathematical prose, so that it is accurate, clear, and readable. However, just so you
will not feel cheated, we will provide one example (in Section 3.2) of a formalized
theorem proved by demonstrating it is a tautology.
If we are not going to use tautologies in our proof writing, then why have we
spent time studying logic and truth tables in this chapter? There are two primary
reasons. One is that tautologies provide us with rules for valid reasoning that will
undergird our mathematical writing. For example, in Exercise 1.2.4 you showed
that [(p → q) ∧ p] →q is a tautology. This tautology is called the modus ponens,
and it is one of several formalized rules of inference that comprise mathematical
reasoning. If we assume p and that p → q, then the modus ponens rule of inference
says we may therefore conclude q. It is not our purpose here to define all the rules
of inference, though you have seen most of them at some point in this chapter.
Exercise 1.2.4 contains the names of several rules of inference.
The second reason we have studied logic and truth tables is that it will provide
us with approaches to the writing of proofs. In particular, we need to understand
how to dissect mathematical statements (the term is to parse sentences) and see
how they give order and structure to our proofs. We also need to know that we can
sometimes write proofs by exploiting equivalent statements. Furthermore, a basic
understanding of logic can immunize us from making certain fallacies—statements
that are not tautologies and therefore not valid theorems.
In this section, we want to outline the four types of demonstrations you will
employ throughout this text when you are asked to compose a proof of a mathematical
proposition. Know that you will always use one of these methods of attack
in any demonstration. Knowing which to choose depends on what your task is,
and is illumined by experience.
1.5.1 Direct Proof
Most theorems are proved directly, which means that the statement of the theorem
and its proof fit into the following template.
Theorem 1.5.1 (Sample). If p, then q.
Proof. Suppose p. Then .... Thus q.