Advice for Writing Proofs

Say what you mean and mean what you say.

Be kind to your instructor by submitting work that is easy to read. In particular, there should be a clear structure to what you write. For instance, what you write should be expressed in complete sentences. Note that a mathematical equation like x + x = 2x qualifies as a complete sentence. For, if you read the notation, x + x = 2, properly, then you get the following complete sentence: "x plus x is equal to 1".

Turn in only well-organized work. Be sure that your work flows unambiguously from beginning to end; the reader should never be confused as to what he or she is to read next.

Use complete, grammatically correct sentences. In particular, correctly use punctuation and capitalization.

Be as concise as possible.

Do not bluff. It is better to assert that "A step is missing here.", (or even better, explain in more detail), than to leave a gap (without explanation). Critique your own work.

Make sure that your proof is not "upside down". If your last line is something like "0 = 0", it is likely that you started by assuming the assertion which you were supposed to prove. This is what we mean by having an "upside down" or "backwards proof". This nomenclature is actually misleading, because it suggests that an "upside down" or "backwards proof" is a proof, which it is not.

Make sure that your proof is not "upside down". Determine the hypotheses of the statement that you are trying to prove. If you are using the direct method of proof of a conditional statement, then this is where you proof should start, not where it ends.

Make sure that your proof is not "upside down". Determine the conclusions of the statement that you are trying to prove. If you are using the direct method of proof of a conditional statement, then this is where you proof should end, not where it starts.

Make sure that your proof is not "upside down". Start with the hypotheses of the statement that you are trying to prove and any relevant facts that you are going to use to reach the conclusion of the statement that you are trying to prove.

Make sure that your proof is not "upside down". Never start with the conclusion of the statement that you are trying to prove.

Focus on just exactly what it is you are trying to prove. Determine with what you are starting. In other words, determine what are the relevant hypotheses and other facts that you are going to use to reach the desired conclusion. Determine to where you are trying to go. In other words, determine what is the desired conclusion.

Focus on how you are going to do your proof. Formulate a strategy. Determine how you are going to use the hypotheses and other facts to get to the desired conclusion. In other words, formulate a strategy to get from your starting point to your ending point.

Define symbols before you use them.

For each assertion in your proof, there should be no ambiguity regarding whether the assertion is known or yet to be proved.

If you have not succeeded in proving what you have been asked to prove, but have partial results, state and prove these clearly.