Notes on Good Proof-Writing

• Be aware that this is a sequential process. That is, if you are asked to prove something, use only results that came before. For example, if you cite Theorem A to prove Theorem B, but you had previously cited Theorem B to prove Theorem A, you have proved nothing. That is called circular reasoning.
• Be aware that the purpose of homework problems is to exercise the concepts covered in a section. So, if you are being asked to prove part of a theorem, citing the theorem accomplishes nothing.
• Imagine that a skeptic is reading all of your work and does not want to accept what you have written. The safe thing to do is to write so well, completely, and correctly that s/he has no chance to criticize what you have done.
• Some "solutions" that are offered are so incomplete that they might merely be considered a hint at a complete solution. You might keep this in mind -- does what you have written more closely resemble a hint or a complete and correct solution?
• Proofs in class will not always be written in the detail we will expect of you, because it can be very time consuming to write every last word and our time is better spent discussing the concepts. You can look to the text, however, to see excellent examples of how proofs should be written. I will also try to post examples of your well-written solutions to many of the homework problems.
• Along these lines, it is a very good idea to indicate what type of proof you are using, at the beginning of your argument, even though the authors do not always do this. If you do not, beware that the "skeptics" (reader, TA, instructor) may have trouble understanding what you are doing, and this could result in loss of credit.
• A written proof should present itself -- it should be very clear about what is going on in terms of the type of proof that will be used, what is assumed, etc.
• It should be a clear stepwise collection of logical inferences, with no "flying leaps" in logic, and in which each step follows clearly from the previous step.
• A clear reason should be given at each step. [We will often relax this requirement and ask only for reasons that are clearly within what I have called the "context" or "focus" of the proof.]
• The conclusion should be clear, as well as why it can be drawn.
• The proof should use correct spelling and proper grammar, and it should be well organized.