0.3: Proof Do's and Dont's
- Page ID
- 22032
This page is a draft and is under active development.
Do's:
- Write the statement to be proved. It should be clear what you are proving.
- Clearly mark the beginning of your proof with the word "Proof".
- Make your proof self contained. In particular, identify all variables used in your proof in the body of your proof.
- Write proof in complete English sentences.
Example \(\PageIndex{1}\): Acceptable
Proof: Let \(n\in \mathbb{Z}\). Assume \(n\) is an even integer. Then \(n=2k,\) for some \(k \in \mathbb{Z}\).
Example \(\PageIndex{2}\): Unacceptable
Proof: \(n\) is even \(\implies\) 2k.
5. Indicate what method of proof you are using. (The default assumption is that it is a direct proof).
6. Learn the definitions and how they come into play when proving various types of statements.
Don'ts:
- Argue from examples. A general statement can't be proved true by showing it is true for special cases.
- Use the same letter to mean two different things within a proof.
Example \(\PageIndex{3}\):
Proof: Let \(n\in \mathbb{Z}\). Assume \(n\) is an even integer. Then \(n=2k,\) for some \(k \in \mathbb{Z}\). So \(n^2=4K^2=2(2k^2)\). Thus \(n^2=2k, k \in \mathbb{Z}\) is even. (The reader asks does \(n=n^2\)?)
3. Assume what you are trying to prove. This is also known as begging the question. You can do it inadvertently in the middle of proof if you are not careful.