Proof Guidelines.
How to write proofs. (Proof in higher
mathematics)
Prof. Warren Esty, Montana State
University
Definition 1: A proof of a result is a sequence
of
statements which demonstrates that the result is a logical
consequence
of prior results.
A proof arranges selected
prior
results into a form that assures that the new result follows
logically
from them. When you want to prove something, which results are
prior?
Definition 2: Imagine
an
ordered list of true mathematical results, one after another. To
prove
any given result, only results earlier on the list, which we call prior
results, may be used. Prior results consist of axioms,
definitions
(which include notations), and previously proven results.
Use scratch paper.
Write down
ideas and definitions on scratch paper without expecting that what
you
write will be the
final organization. After you have written enough that you can see
the
final proof, create a final copy by selecting and organizing
the
useful parts from all that you have written.
(All your steps must be prior
results. If you have a good idea and it is true, it does not belong
in
a proof unless it is already known to be true as a prior
result you can cite.)
Proofs of
generalizations.
Many theorems are conditionals. If it is a
conditional,
Analyze the
conclusion.
Translate
new terms in the conclusion into more-primitive terms (using
definitions in sentence-form)
"Translate, work
with the
older terms, translate back."
If the translated form fits one of our
logical equivalences, it may be proved in the alternative form, and
often is.
If
the conclusion is itself a conditional, use "A Hypothesis in the
Conclusion":
"H => (A
=> B)" is logically equivalent to "(A and H) => B" where "A"
is intentionally given first. Usually, beginning with "A" is a good
idea.
If a direct proof does not seem to work,
try using the contrapositive.
If you don't have an idea about how to do the
proof,
start in the place
suggested by our logical equivalences and see where that takes you.
If it does not seem to be working, the conjecture
may be false.
Begin with a general,
arbitrary case (not a constructed case).
Look for prior results that might be similar and helpful.
If the variable is integer-valued, induction might work
Memorize and use definitions in sentence-form. (Because
proofs are sequences of sentences.)
Existence
statements are usually proved in two stages:
1) Exhibit a candidate, and
2) Show it has the desired properties.
About a conjecture: Decide if it is true. Try
examples to test its truth. Look for
possible
counterexamples. If you can't find one, then seek a proof.
If it is true, prove it.
If it is false, give a particular
counterexample.
Counterexamples should assign particular values to all of
the
letters.
Simpler counterexamples
are better.
(because they are more
memorable and more
likely to illustrate the key idea without unnecessary
complications.)
When you are stuck:
Use the hypotheses. If a conditional is
true it is likely you will need to use all the hypotheses.
Review the definitions. Often the key is
in knowing precisely and then using the definition of some term.
If you try a proof and can't complete a
step, look
for a counterexample in which that step is false.
If you mean to assert something exists, mention existence
explicitly.
Uniqueness proofs: Give the thing(s) two names and show they
must be two names for the same thing.
Advice for harder problems:
If the problem is abstract,
construct a simple example (emphasis on
"simple") to make sure you know what the problem says.
Also, construct simple examples of
immediately prior results to make sure you know what they say.
To prove a result
you must assemble prior results, so you must know
the prior results. That is,
know the formal definitions of the new terms in
the
result, and
be familiar with related previous results. (For
example, often the proof of Theorem 10 uses Theorem 9.)
Here is a page on some things to
avoid in proofs.
Return the the main page of Math
242 or of Math 381.