COT 3100 sec. 7094X, Fall 1999
Supplementary Notes on
How to Write Good Proofs
Why is this important?
Some students may not at first understand why it’s important for their career that they learn how to
write good mathematical proofs. They might say, “I understand the basic concepts and properties
of discrete structures, and I know how to work with them, why is it important that I be able to
prove things about them?”
The reason is that proof-writing teaches not only logical reasoning skills, but also the critically
important, very general skill of knowing how to clearly communicate a logical argument.
pose you are working at a company, and you need to explain to a co-worker, client, or supervisor
the reason why you recommend a particular algorithm, procedure, strategy, or course of action. If
you are able to give a clearly stated, concise, coherent logical argument in support of your pro-
posal, your chances of successfully convincing your audience of the merits of your proposal (and
of your own competence!) will be greatly improved.
The ability to communicate effectively
was recently rated
most important job skill by industries that hire our graduates!
You can look at writing mathematical proofs as simply an exercise in clearly and convincingly
communicating a logical argument.
The domain of our proofs happens to be the discrete mathe-
matical structures that we have been studying, but many of the same skills (of clear thought and
clear communication) that you learn when doing mathmatical proofs can carry over to improve
your skills at composing persuasive rational arguments in just about any area of endeavor.
And of course, knowing how to read and write proofs (and rational arguments in general) should
be very helpful to you for your later classes in this degree program, and it will be absolutely
essential if you plan to go on to graduate school to eventually teach or do original research in any
area of engineering, science, computer science, or mathmatics.
Law and business graduate pro-
grams also require the ability to present clear arguments.
2. What is a proof?
A proof is merely an argument that firmly establishes the validity of a statement.
Ideally, the most effective means of proof is an interactive discussion.
Suppose you are trying to
convince someone of the truth of a statement A.
They don’t at first understand why A is true, so
you give them an intermediate statement B in support of A. They may then see why B is true, but
not yet why B implies A.
So you give them an additional intermediate step C.
Now, maybe they