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Proof Tips
•
Be very familiar with the different methods of proof, but do not necessarily begin
immediately to try to figure out which one to use (that is, don't panic!).
•
Think "direct proof," unless observations suggest otherwise. For example, a direct attack
may not suggest much, but considering ~S seems important, or ~S seems clearly false.
Or, perhaps a direct proof looks difficult or messy and considering ~Q => ~P looks more
fruitful. In particular, some proofs inherently involve a "not," such as trying to show that
something is not true or trying to prove a number is irrational (defined as not rational).
Contradiction and contraposition sometimes work very well in such a situation. On the
other hand, some people become very enamored with proofs by contradiction and
contraposition and use them when a direct proof is simpler and much clearer, so be
careful.
•
Keep in mind what you know (what you are allowed to assume), but consider carefully
what you need to prove and what must be shown to prove it
, and proceed in a deliberate
fashion. That is, you must be aware of how to prove what you need to prove, this must
direct the way you go about your proof, but beginning by using what you know may put
you in a hole from which you cannot recover. So, consider carefully what you must
demonstrate in order to prove what you wish to prove, and let that direct how you use
what you know.
•
Do not feel that everything must flow naturally from the start. That is, you might not
necessarily see immediately how to get to your conclusion from what is given and what
you know. Instead, as you engage in trial and error, pieces of the puzzle start to become
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This note was uploaded on 03/18/2012 for the course MAT 108 taught by Professor Staff during the Winter '10 term at UC Davis.
 Winter '10
 Staff

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