I’m teaching Discrete Math this semester. Discrete Math is our college’s “introduction to proofs” class. We spend a lot of time talking about and practicing proofwriting. In earlier blog posts I shared my “Nuts and Bolts of Writing Mathematics” and an “editing checklist” that I give to them.

Yesterday, I gave them an example of a very, very poorly written proof that I created. (It was fun trying to insert as many problems as I could into the short proof!) I had them work on the assignment in class for about 7 minutes. Then we came together and talked about all the errors that we could find, and finally we rewrote the proof in the correct way. Here’s a pdf of the assignment, and I’ve posted a screenshot below.

In case you are wondering, here are the errors in the proof. Numbers refer to sentence numbers.

- The proof should be begin immediately after the word PROOF.
- The entire proof should be written in paragraph form.
- 1. Capitalize the first word of the sentence.
- 1. Split this sentence in two: The first sentence should be “Let
*m*be an odd integer and*n*be an even integer.” In the second sentence we apply the definition of even and odd. - 1. Do not use
*k*for both the even and the odd numbers. - 1. Don’t use a symbol (#) in place of a word.
- 2. Start the sentence with a word, not a mathematical expression.
- 2. The correct expression is
*mn*^{2}, not*m*^{2}*n*. - 2. “Equivalent to” is a term we use for logical expressions, not numbers. It should be “equal to”—and in fact, we should just use “=.”
- 2. Algebra error: the term 8
*k*should be 8*k*^{2}. - 2. In a 200-level math class we do not have to show all of these algebraic details.
- 3. This is not how we show that a number is even—we must show that
*mn*^{2}satisfies the definition of even. - 3. Do not use the word “obviously.”
- 3. In a proof like this we’d have to show that the three terms satisfy the definition of even.
- 3. “There” should be “their.”
- 3. Missing dollar signs around 2
*k*in the LaTeX code. - 4. Do not use the passive voice.
- 4. Missing period at the end of the sentence.
- 4. and 5. Write in present tense.
- 5. Do not put an example in a proof.
- 5. There are missing dollar signs in the LaTeX : after
*m*=5 and before*n*=4. - 5. It should be first person plural, not first person singular (“we” not “I”).
- 5. Replace “is = to” with “=.”

Here is our corrected proof: