A Guide to Writing Proofs

1. A formal proof is a series of statements. Each statement must either be a hypothesis (assumption) or follow from previous statements in the proof and previously proven theorems. The proof must end with a statement that is either the intended conclusion or that trivially implies the intended conclusion. This is not just a matter of style, it is the nature of logic.

2. An exposition of a proof is a piece of writing that convinces a reader that a formal proof exists. It has both less and more than a formal proof. Detailed, obvious steps are left out. But be sure that they really are obvious to your audience. An exposition of a proof mixes in human language statements to make the proof more readable and to make clear what follows from what.

3. A series of mathematical equations with no English to connect them is not acceptable. Do not write

equation1

equation 2

equation 3

Rather write something like

We have equation 1 from the hypotheses. By taking x= 9, this implies equation 2. By the division theorem, we then have equation 3.

4. On the other hand, do not say too much. Give only the facts that are in the direct chain of implications from the hypotheses to the conclusion. Do not restate a fact 2 or more ways because you think it's a nice fact. State only what is really needed. Do not give examples unless that's what's asked for. Irrelevant information only obfuscates what is being said. It makes it hard for the reader to follow because she cannot figure out what the statements of the proof follow from. Notice that in texts, this sort of stuff goes in the part between a proof and the next theorem.

5. Do not begin a proof by stating the conclusion of the theorem. This is completely confusing. My standard assumption is that if you make any plain statement in a proof, you think that the statements that come before it imply it. If you want to be able to refer to it somewhere, write something like "We want to prove that equation 1". Or simple leave it out altogether, and be sure the proof ends with the conclusion. Or restate the entire problem.

6. Say what kind of proof you are writing. Direct proof, proof by contradiction, proof by induction. This helps the reader tremendously to know what to expect.

7. An example is not a proof. A picture is not a proof. 5 < 6 does not imply that for all x and y, x < y. Occasionally a proof is made clearer by a picture or an example. But they are not part of the proof, they are just for clarity. A picture or an example does not imply anything (well, a counterexample can imply something is false, but that's the only way an example proves something).

8. Look at the style in the homework solutions I give you and try to follow it in your writing (or if you think it could be made clearer, develop a clearer style).