# Plural for math notations

I have a question about the usage of plural in mathematics in the following context.

Suppose we have functions f1, f2, ⋯.

Which of these sentences is correct (if any)?

1. Among the functions fi's there is no repetition.
2. Among the functions fi there is no repetition.
3. Among the functions fi's there are no repetitions.
4. Among the functions fi there are no repetitions.

The focus is on whether I should use 's or not. Also, is or are.

• People may downvote this question since it is about a mathematical convention and not standard English, but I have given my opinion below. – GoldenGremlin Jul 14 '16 at 0:20
• I hope it is not down-voted, because it is a good example of proper use of mathematics in English, and so often the two seem to clash. – KWinker Jul 14 '16 at 2:16

You can think of "fi" as shorthand for the list of individual functions "f1, f2, ..., fn, ..." Understood as such, the correct sentence would be:

Among the functions fi there are no repetitions.

This would be a shorthand for:

Among the functions f1, f2, ..., fn, ... there are no repetitions.

Notice the plural "functions" and the use of the plural verb "are".

Alternatively (and maybe less standardly), you could think of "fi" as a kind of count noun which applies to every function in the list. Understood as such, the correct sentence would be:

Among the fi's there are no repetitions.

Normally, you would not use an apostrophe in a pluralization. But you can get away with using the apostrophe here (even though it is not indicating possession), just like you can get away with using it to make a pluralization for something like a letter, as in:

There are two a's in the word actual.

The apostrophe is needed to forestall confusion.

I don't think there is hard and fast rule about which is the right way to view the mathematical convention of using "fi". Your best bet is to pick one way of understanding it and stick to it. Or else forget about it entirely and just say:

Among the functions there are no repetitions.

After all, why muddy the page with needless formalism?

• I find the use of "the fi's" jarring and recommend against it. But I'm upvoting because everything else in this answer is excellent advice. – David K Jul 14 '16 at 0:56