## Proofs

A proof is an argument used to show that something is either unambiguously true or unambiguously false.

• It should be written in a concise form.

• Arguments should be clear and consistant.

• A proof should not start by assuming the claim is true.

• A proof should have sufficient explanation of why certain steps are taken.

• There should be no room for confusion in the proof.

• We can use certain rules from both the lectures and textbook without additional proof.

• There is no need to over-complicate a proof by including a number of mathematical systems to make it shorter.

• Often negative numbers or $0$ may be an issue with your proof.

• Ensure that you prove your statement is true for all numbers that it allows.

### Choosing a Proof Method

There is no clear answer to the question of which proof method should be chosen.

• You can try every method until something works.
• Experience/comparing with problems you've seen before helps a lot.
• Often there are several approaches that work.

$$\begin{array}{c|c} \text{Claim Type}&\text{Possible Method}\\ \hline\\ \text{If/Then}&\text{Direct Proof}\\ \text{Modular Arithmetic}&\text{Cases}\\ \text{Something Exists}&\text{Construction}\\ \text{Sequences/Recursion}&\text{Induction}\\ \text{Stuck?}&\text{Try Contradiction}\\ \text{Still Stuck?}&\text{Maybe it's false, try disprove}\\ \end{array}$$