Proof-Writing Tips
In addition to the tips below, see
Chapter 0 of
Mathematical Proofs, Third Edition by Chartrand, Polimeni, and Zhang
- There's a formal or ritualistic structure to mathematic proofs.
Begin with "Proof:" and mark the end of your proof with "QED",
a box, or some other symbol. QED is from Latin phrase quod erat demonstrandum,
meaning "which was to be demonstrated".
- Note: if you are doing a proof by contraposition, by contradiction, by induction, or by complete induction,
start with "Proof by ..." instead of just "Proof". This alerts the reader to what s/he should expect.
- Think about the type of proof you are doing and its main steps. What are its assumptions? Make sure to
state them first.
- Definitions are critical in connecting one concept to another. Use them. You do not necessarily
have to rewrite the entire definition within your proof, just refer to it.
- Having trouble getting started on a proof? Write out its main steps and the definitions of the
terms involved. How can you use those definitions? Work on scrap paper first.
- Keep symbols to a minimum in written work (they are great for presentations at the board).
Avoid using symbols at the beginning of a sentence.
- Use proper English grammar in all sentences.
- Justify each step in the proof.
- Use new paragraphs for each case in a proof or for each "direction" in an if and only if proof.
Indicate which case or direction is being considered.
- Define each variable before you use it. For example, you should state
"Let \(n\) be a positive even integer," before you begin using \(n\).
- Use words carefully. Distinguish carefully between moments when you want to use "since"
or "because", and
moments when you want to use "if"; similarly for "and" and "implies" or
"then".
- When you doing an set arguments especially with unions it is often necessary to examine two separate
cases. Follw the first case through completely before you considering the second. Try not to do too
much at once.
- End your proof by saying what you have proved. (This should be what you set out to prove!)
If you were
trying to prove that the sum of two odd numbers is even, then your last sentence of your proof should be
something like: "Therefore, the sum of odd even numbers is even."
- If you are doing a proof by induction or complete induction, be sure that your
conclusion says this, for example, "By the Principle of Mathematical Induction,..."
Presentation Tips
- Review the "Proof Writing Tips" above to make sure your work is ready to present.
- Introduce yourself.
- Begin your presentation with a short verbal summary; mention key theorems and definitions that will be used, before you
start writing the proof on the board.
- Write clearly and big enough for the class to see, but not too big!
- Keep the board organized. Split it visually in half or thirds as appropriate.
- Likewise, speak clearly and loud enough for everyone to hear you. Speak as you write, but try
to avoid talking to the blackboard when possible.
- Proofs written on the board should stand on their own
even if you weren't talking (but you will be). Don't omit
details of your written proof just because you are speaking the words.
- Write full sentences.
Pause after each sentence or two to be sure that your classmates are following you.
- Try to avoid blocking what you are writing. This is hard, but you can take advantage of
your pauses to get out of the way. Write with one hand behind your back to get that arm out of sight-lines.
- Your classmates are your audience. Be sure to look at them;
try not to present to the professor.
- Be enthusiastic! Enjoy yourself. If you enjoy it, so will the audience.
These tips are based on those of my colleagues: Professor King's Tips
and Professor Orr's Tips.