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Consider a typical mathematical sentence defining two tuples: (s_i)_{i=1}^n and (t_i)_{i=1}^n:

Let (s_i)_{i=1}^n = X (,) and (t_i)_{i=1}^n = Y.

The parens around the comma mean that it's unclear whether the comma actually belongs there.

Is the comma before “and”

  1. necessary,

  2. forbidden,

  3. optional without a change in the meaning, or

  4. optional with a change in the meaning?

I know a rule saying that, in general, the comma is necessary if two independent long sentences are joined by “and” and forbidden if both of them are short (I think, 4 words at most). Because of maths, no idea. It's not even clear to me what the structure of the whole sentence is: two independent clauses with omitted second “let” (“Let (s_i)_{i=1}^n = X (,) and [let] (t_i)_{i=1}^n = Y.“) or one single clause with two objects.

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  • Math notation does not contain sentences. I would remove the period in your statement.
    – Lambie
    Dec 3, 2021 at 16:16
  • @Lambie Note: We are not speaking about standalone formulas (where you omit the period in certain circumstances), but about a sentence. Every sentence (with lots of maths or without it) starts with a captial letter and terminates with a full stop.
    – user142975
    Dec 3, 2021 at 16:18
  • Math equations are not sentences. They are expressions. en.wikipedia.org/wiki/Let_expression
    – Lambie
    Dec 3, 2021 at 16:34
  • @Lambie Yes and there are two expressions embedded in a sentence. This sentence is in the middle of a paragraph; there is text before and after the sentence. The full stop separates this sentence from the next one. Anyway, I'm not here to explain you all these trivialities about the full stop; learn this yourself.
    – user142975
    Dec 3, 2021 at 16:38
  • Let X [be whatever] and Y [ bewhatever]. No comma, especially not in parenthesis. It is just a complex sentence.
    – Lambie
    Dec 3, 2021 at 16:43

2 Answers 2

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Mathematician here.

Short answer. In written mathematics the comma would be omitted.

Longer answer. I would rewrite the sentence (and it is a sentence) this way, with more white space and less formal notation, to make it as easy as possible for the reader to parse:

Let

X = (s_1, ..., s_n)

and

Y = (t_1, ... , t_n).

The variables being defined (X and Y here) should be the left members of the defining expressions. That's better English usage and what you would write in a program.

Some mathematical typographical style guides would omit the period at the end of the second displayed equation. I always include it.

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  • No space for block equations. On the contrary, we save space whenever we can. By the way, we define the tuple components s_i and t_i (1⩽i⩽n), and NOT X and Y. I think I was extremely clear on what is being defined, wasn't I? The variables X and Y come from elsewhere; we already know (from prior text omitted in the example for simplicity) that their sort is an n-tuple sort.
    – user142975
    Dec 7, 2021 at 0:18
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My math(s) knowledge is not enough to say whether this is a linguistic sentence that contains two equations, or simply another sort of written expression. What if it were a sentence made entirely of words, though? Its structure would be analogous to:

Let Ted be the team coach and Beard be his assistant.

In this construction it's pretty clear to say that option 3 is true, though I would go so far as to say that a comma is "optional but not advisable." The comma is absolutely not needed. You could add it to demarcate ideas, but I wouldn't do so with words, and I would find it even more potentially confusing when dealing with mathematical symbols.

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  • If you actually had “be” in writing in my example, I'd fully agree with you concerning the absence of the comma.
    – user142975
    Dec 3, 2021 at 16:49
  • I didn't spell this out, but I was thinking that part of the issue with parsing the math-symbols bits into language is that the equals signs are the "verbs" of these clauses. Read aloud, I honestly have no idea what some of it comes out as, but I assume it's "Let ___ equal X." If we are bound by the forces of grammar, then they apply here the same as in my example. (Also, I would hope you have some kind of specialized style guide! When I wrote papers about music for my university there were definitely specific rules about usage around musical terms and symbols that the university spelled out. Dec 3, 2021 at 16:57

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