# Proof Writing and Presentation Tips Fall 2018

Professor: Erika L.C. King
Email: eking@hws.edu
Office: Lansing 304
Phone: (315) 781-3355

### Tips for writing good proofs

1. Begin with "Proof:" and mark the end of your proof with "Q.E.D." (which stands for "quod erat demonstrandum", Latin for "that which had to be demonstrated", $\Box$, or some other symbol.
2. If you are doing a proof by contraposition, by contradiction, by induction or by complete induction, start with "Proof by ..." instead of just "Proof".
3. Think about what type of proof you are doing and what the two main steps are. What assumptions should you state first?
4. Avoid beginning sentences with symbols; keep symbols to a minimum in general (in written work -- they are good for presentations where they will be accompanied by spoken word).
5. Use full sentences with proper punctuation (this includes capitalizing the first word in every sentence and using periods at the ends of sentences).
6. Do not abbreviate words.
7. Justify each step.
8. Use separate paragraphs for each case/direction and make it clear which case/direction it is.
9. Define your variables before you use them. For example, say "Let $x$ be a real number greater than two." before you begin using $x$.
10. Remember that definitions are a key in connecting one idea to another. Use them. This does not necessarily mean that you need to rewrite the entire definition within your proof, just refer to it.
11. Use words carefully. Distinguish carefully between moments when you want to use "since" and moments when you want to use "if"; similarly for "and" and "implies" or "then"; and also for "the" and "a".
12. When you are doing an elemental argument involving unions it is often necessary to look at two separate cases, following the first one through completely before you look at the second. Be careful not to try to do too much at once.
13. Always end your proof by saying that you have proved what you set out to prove. For example, if you were trying to prove that the sum of two even numbers is even, then your last sentence of your proof should be something like: "Therefore, the sum of two even numbers is even."
14. If you are doing a proof by induction or complete induction, that means that you were able to prove what you proved by the Principle of Mathematical Induction or by the Principle of Complete Induction. Be sure that your conclusion mentions this with something like "By the Principle of Mathematical Induction,...".

### Tips for giving good proof presentations

1. Read above to make sure the proof you are about to present is ready to go.
2. Begin with a verbal short summary, mentioning key theorems and definitions that will be used, before you start writing the proof on the board.
3. Write clearly and big enough for the class to see, but not too big!
4. Keep the board organized.
5. Write full sentences.
6. Pause after each sentence or two to be sure that the class is following you.
7. Avoid blocking what you are writing. This is sometimes difficult, but you can take advantage of your pauses to get out of the way.
8. Speak clearly and loud enough for everyone to hear you.
9. Your classmates are your audience. Be sure to look at them and not to present to the professor.
10. Be genuinely enthusiastic! This is more important than you might imagine.

Erika L.C. King