Translate to Symbolic Logic. A typical comment made when proofs are attempted is
I do not know where to start!!!!This statement is made with a great gnashing of teeth and wringing of hands. One procedure to follow is comparable to that of solving a problem in basic algebra.
First, translate what you are requested to prove into symbolic logic. Then
seeing the structure of the translated sentence you can select a mode of proof.
Still, knowing a mode of proof that could be used does not guarantee
success. Suppose you want to attempt to prove a sentence of the type
by using the Rule of Conditional Proof. You want to assume
P and deduce Q. A question often asked is
How do I get from P to Q?There is no certain way. No one way will always work. Certainly, knowing to assume P and deduce Q is a step in the right direction. The mode of proof provides the structure for the proof; building this structure is usually a more creative task. I can give a few hints.
Analogy. An important aid in carrying out proofs is to get ideas from other proofs. This is supposed by comments of mathematicians who argue that to be good at mathematics you need lots of practice; lots of exposure to different proofs.
Analytic Process. This is known as working backwards. You want to
prove . Start with Q and try to find an R such that
. Then try to find and S such that 
. Then look to
see if 
. If not, try to fill in another step. Continue this until
you find a sentence 
such that 
and 
Do not be surprised if you do not
see this process outlined in a text or reference book. It is rare that if this
processed is used it is then explicitly mentioned. Usually the proof will be
given as
Do-Something Approach. This is simply trial-and-error. You want to prove
by assuming P and deducing Q. You have no particular way to
get from P to Q; but start out, get involved, do something, try different
approaches, prove all that you can. You do not have to show all of this in
your final version of your proof, but it can help you get started. When
reading proofs in mathematics texts and journals, you are not aware of the
blind alleys and unsuccessful attempts preceding a successful proof. This
leads you to think the established mathematician never follows a wrong path
or makes a mistake. Trial and error is very much a part of mathematical
creativity.
Use of Definitions. Another helpful procedure is to recall all relevant definitions. It is a tendency to read a definition and ignore its importance in later proofs.
Use of Previously Proved Theorems. It is helpful--nay, it is essential in starting a proof to examine all previously proved theorems for results which might be relevant to the proof.