Writing Better Proofs Part 3: Notation

When writing a proof we often have a choice of notation. Perhaps we are naming variables or deciding if functions act from the left \( \theta(x) \), or right \( (x)\vartheta\). We may be referring to existing theorems or constructions (is it the HOMFLY polynomial or the HOMFLYPT?) or may need to introduce our own.

In all cases there is a single golden rule: be consistent.

As long as you set out your notation early and stick to it, you are being clear. Change notation half way through or be vague about it, and you are just going to lose your reader.

However, there are a few “silver rules” you should also follow.

Use Existing Standards

If you are answering a question or proving a lemma or statement, you should always use the notation set out in the question. If the current literature favours one standard of notation, unless you have very good reason not to, use that notation in your work.

The only reason to break this rule is if your notation is clearer or conveys more specific information. If so, introduce it clearly at the outset and stick to it.

Problem Prove that any map from \(M\) to a filtered module \(N\) is surjective.

Solution Let \(N\) be filtered by \(N = N_0 \supset N_1\supset\ldots \supset N_m\). Then let \(f : M \to N_0\) be any map. We will show that \(f\) is an epimorphism and hence surjective.

…

Perhaps the most common example of this is the \((\epsilon,\delta)\) proof-type in analysis. Here, almost always, \(\epsilon\) is fixed before \(\delta\). Please do not start a proof For each \(\delta\) there exists an \(\epsilon\) unless you are making a joke¹.

Choose Variable Names Carefully

Most often (in English proof writing) certain variable names are tacitly “reserved” for certain uses. In particular, variables \(i\) and \(j\) are often indices that are summed over, \(n\) and \(m\) are likely fixed. The variables \(x\) and \(y\) are unknowns and \(p\) is often a prime.

These are to be taken as guidelines only (nobody will complain too loudly if you sum over \(n\) or if \(x\) is fixed) but breaking these conventions can lead to confusion.

Consider the equation \[i = \sum_{n = 0}^j n\cdot (m_x +j) \] versus the semantically equivalent \[x = \sum_{i = 0}^n i\cdot (m_j +y). \]

The latter is far easier to read because the reader is expecting the various elements to go where they do. Not being surprising to the reader is a key element in writing a good proof.

Be Aware of How Variables Look

Some variables look very similar, especially if handwritten. Consider the notation below.

Let \(a_0, a_1,\ldots, a_n\) be chosen such that \[ a_0 + a_1\alpha + \cdots a_n\alpha^n = 0.\]

Here the alpha’s and a’s are just begging to be mixed up. Best to pick different coefficient names².

On the other hand, the letters \(i\) and \(j\) are also easy to mix up. If reasonable, other “summation” variables such as \(r\) and \(s\) could be used. Otherwise be careful to make sure it is clear which is which. Avoid the dreaded \(i_j\) and \(j_i\) at all costs.

Be Sparing with \(\Sigma\) and \(\Pi\)

The two capital Greek letters \(\Sigma\) and \(\Pi\) have special meaning in most subjects. They represent summation and products respectively.

Having said that, sometimes they can be used as variable names. Its status as a “super capital S” means that \(\Sigma\) is often used for a set of sets. The letter \(\Pi\) sometimes is used to contain a set of indices or denote a particularly special prime ideal.

LaTeX sets these apart by adjusting their aspect ratios very subtly. The string $\sum \Sigma \prod \Pi$ renders as \(\sum \Sigma \prod \Pi\). Sometimes this is sufficient to hint to the reader the difference between the symbols, but when handwritten this is almost impossible.

Don’t let any of this dissuade you from coming up with your own notation, though. As long as it is clear and consistent, you are in the right. It may be better than the existing notation and may make your life substantially easier. Perhaps new notation allows you to distinguish between two subtly different situations, or simplifies your equations from a horrible case statement to a simple summation.

Just remember the golden rule: be consistent.

This is part of a series on writing better proofs. The pages in the series are not static and will be updated and improved as time goes on. If you have comments or suggestions, please email me. I’d be particularly interested in hearing if you disagree with me or have suggestions for topics to cover.