What is a correct and popular format (punctuation, conjunction, period) to write a list of equations?

Question

Which of the following ways of writing equations is acceptable and/or preferable in mathematical typography?

Format 1

The four equations in the computation graph are: \begin{align*} a_1 & = f(W_1 a_0 + b_1). \\ a_2 & = f(W_2 a_1 + b_2). \\ a_3 & = f(W_3 a_2 + b_3). \\ a_4 & = f(W_4 a_3 + b_4). \end{align*}

Format 2

The four equations in the computation graph are \begin{align*} a_1 & = f(W_1 a_0 + b_1), \\ a_2 & = f(W_2 a_1 + b_2), \\ a_3 & = f(W_3 a_2 + b_3), \\ \text{and } a_4 & = f(W_4 a_3 + b_4). \end{align*}

Format 3

The four equations in the computation graph are \begin{align*} a_1 & = f(W_1 a_0 + b_1), \\ a_2 & = f(W_2 a_1 + b_2), \\ a_3 & = f(W_3 a_2 + b_3), \\ a_4 & = f(W_4 a_3 + b_4). \end{align*}

Format 2 reads like it is closest to being correct English. But I think I have seen Format 3 more often in math books. Is there any consensus on which of these styles is correct and/or appropriate in professional mathematical typesetting?

Note: This may look like a duplicate of Punctuating blocks of equations but this question is different from the other one due to the following reasons:

• The other question has equations ending with digits which makes it difficult for us to end every equation with a comma because the comma may be confused for a decimal point by the readers of some countries. This question does not have equations ending with digits, so it is safe to use comma at the end of every equation.
• The other question has an easy and safer solution (i.e., rewrite the equations as a matrix). However, in this question we have four separate equations and the question about punctuation and conjunction needs to be addressed.

Here are some related posts that I have already read:

• https://mathoverflow.net/a/6683
• https://tex.stackexchange.com/a/170691
• https://english.stackexchange.com/a/22551

Answer 1

I am of the persuasion that one should be definitive about blending notation and English. Therefore, I would recommend or prefer format 2, and I will also recommend my own format.

My Format

I personally do not like English punctuation in mathematical notation. There is already plenty of comma and semicolon usage (e.g., in notating ordered pairs). Instead, I like to include clear spacing. In your example, I opine that indentation is enough.

The four equations in the computation graph are \begin{align*} a_1 & = f(W_1 a_0 + b_1) \\ a_2 & = f(W_2 a_1 + b_2) \\ a_3 & = f(W_3 a_2 + b_3) \\ a_4 & = f(W_4 a_3 + b_4) \end{align*}

I find that the colon interrupts the flow of reading too much.

I would also like this format:

The four equations in the computation graph are $$\left\{ \begin{align*} a_1 & = f(W_1 a_0 + b_1) \\ a_2 & = f(W_2 a_1 + b_2) \\ a_3 & = f(W_3 a_2 + b_3) \\ a_4 & = f(W_4 a_3 + b_4) \end{align*} \right.$$

I have seen the brace on the left plenty of times, usually to group simultaneous equations when the system is not represented as an equality of two matrices.

Format 2

If it bothers you to not end a sentence with a period (it bothers me), and you like to use proper punctuation throughout your writing, format 2 is most appropriate.

The four equations in the computation graph are \begin{align*} a_1 & = f(W_1 a_0 + b_1), \\ a_2 & = f(W_2 a_1 + b_2), \\ a_3 & = f(W_3 a_2 + b_3), \\ \text{and } a_4 & = f(W_4 a_3 + b_4). \end{align*}

As an alternative, which parallels standard alignment to the left margin of documents, consider

The four equations in the computation graph are \begin{align*} a_1 & = f(W_1 a_0 + b_1), \\ a_2 & = f(W_2 a_1 + b_2), \\ a_3 & = f(W_3 a_2 + b_3),\text{ and} \\ a_4 & = f(W_4 a_3 + b_4). \end{align*}

Style for English

When publications are written in pure English or items in a list are more than simple noun phrases, I see this bullet format most often:

Your choices are

• option A;
• option B;
• option C; and
• option D.

This is what APA recommends and how legal documents are formatted (though those use letters and numbers instead of bullets).

You could try preceding the equations with bullets, but that is not common in mathematical publications.

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Also relevant is this question, on which I have also posted an answer.