Writing Proofs

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Writing Proofs

Based on “Writing Guidelines for Mathematics Proofs” in section 1.2 of the textbook, and the “Minimalist Introduction to LaTeX.”

Example with LaTeX

There are some generally well-written, up to occasional and inevitable typo-level errors, examples in the discussion. (Even though I know that typo-level errors are “inevitable,” a well-written proof should be proofread, follow conventions of grammar and spelling, etc.)

Let’s try an example that’s not in the discussion, namely proving that the sum of a rational number and an integer is another rational.

We wrote the proof in LaTeX, developing proof ideas alongside talking about the proof-writing rules and LaTeX features we used.

The main idea of the proof is to use the definition of “rational number,” namely a number that can be written as a fraction a/b where a and b are integers and b ≠ 0, to expand out what a rational plus an integer would look like, and then, by putting that sum over a common denominator, show that it is also a ratio of 2 integers with a non-zero denominator.

Both the LaTeX source file that we wrote, and the resulting PDF output file, are available from Canvas.

Some guidelines for writing proofs that this one uses include

• Starting with a sentence that says what the proof will assume, and what it will show
• Ending with a summary of the theorem proven (which often looks very much like the theorem statement, but that’s OK) and an end-of-proof phrase or marker (conveniently inserted by LaTeX, so we didn’t have to worry about it explicitly)
• Attention to mathematical typography conventions, including writing variable names in italics, and making large or important equations displayed (i.e., set off from surrounding text and centered)  instead of embedded in regular sentences.
• Thinking consciously about the audience who would read the proof (a good assumption for proofs in courses is that the audience is other students in that same course) in order to decide how much or how little detail was needed
• Referring to the people writing or reading the proof as “we,” rather than “I” or “you.”

Some new (i.e., not in the “Minimalist Introduction to LaTex”) LaTeX features that we used to write the proof include

• The “amsthm” package, which provides helpful commands and environments for writing theorems and proofs
• The newtheorem command, which creates environments for stating theorem-like things (e.g., theorems, lemmas, postulates, axioms, conjectures, definitions, …). There are so many such things that rather than LaTeX trying to foresee all of them, it gives authors a way to quickly define the ones they need, via newtheorem, which has as its arguments the name of the environment to create and a word or phrase to label uses with. In our case, we used it to create an environment thm that represents theorem statements; every theorem statement defined in this environment will be labeled with the word “Theorem.”
• The proof environment, which identifies a block of text as being a proof.
• The \ne command, which produces a “not equal” sign
• The \frac command, which produces a fraction written as a numerator over a denominator.

Problem Set

There’s a new problem set available, about proofs and writing them.

See the handout for details.

Next

“Compound” statements, i.e., statements made by joining simpler ones with “and,” “or,” etc.

Please read section 2.1 in the textbook.

Please also contribute to this discussion of compound statements.

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