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This is a question about deciding singular vs. plural verb where the subject contains multiple objects in it. Let me set the context first.

I have a mathematical problem where I need to find a solution to a problem. A single solution is a collection of three functions I need to find. For examples, the functions f(x) = 1, g(x) = 2x + 1 and h(x) = 3x + 2 may form a single solution. Similarly, the functions f(x) = x + 1, g(x) = 2x and h(x) = 0 may form yet another solution.

Now after solving the problem I find that there is only solution to the problem. Which of the following ways is the right way to express this thought?

  1. The functions f(x) = 1, g(x) = 2x and h(x) = x + 2 is the only solution to the given problem.
  2. The functions f(x) = 1, g(x) = 2x, h(x) = x + 2 is the only solution to the given problem.
  3. The functions f(x) = 1, g(x) = 2x and h(x) = x + 2 are the only solutions to the given problem.

Option 3 sounds grammatically correct to me but sounds mathematically incorrect to me because the functions f(x), g(x) and h(x) are not three different solutions I have found to the problem. Those 3 functions together constitute a single solution to the given problem.

So is option 1 or option 2 correct? If none of the options are correct, what is a correct way to express this thought?

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  • 1
    It needs rephrasing. << The only solution to the given problem is when the functions in question are defined as follows: f(x) = 1; g(x) = 2x; h(x) = x + 2 >> (and I'd use bullet-points).
    – Edwin Ashworth
    Commented Apr 13, 2021 at 11:57
  • While rephrasing the sentence is definitely one way, I want to understand if English grammar already accounts for a situation like this. Does English grammar allow anyway to write something similar to one of the three options I presented with minor tweaks? I am trying to understand if there is a way to express "X, Y and Z is ..." where we know from the context that "X, Y and Z" constitute of a single instance of something.
    – Lone Learner
    Commented Apr 13, 2021 at 12:04
  • 2
    Notional agreement is certainly used (at least to some extent) by many if not most proficient Anglophones. 'Bacon and eggs is my favourite meal' / 'Health and safety is our primary concern'. But your suggestions here are at least bordering on the outlandish, and should be rewritten in accordance with Orwell's Sixth (Rule): 'avoid anything sounding outlandish, even if you have to resort to a breech in grammaticality' (a fair paraphrase).
    – Edwin Ashworth
    Commented Apr 13, 2021 at 15:36
  • Apples, oranges and grapes are the fruit to eat. So, the answer is yes. This is not even a math question, really. The functions f(x) = 1, g(x) = 2x and h(x) = x + 2 are the only solutions to the given problem. RIGHT.
    – Lambie
    Commented May 13, 2021 at 20:15
  • The copula is just isn't up to the job. You need a verb with semantic content such as composed of. The only solution is composed of the functions ...
    – Phil Sweet
    Commented Oct 11, 2021 at 2:18

3 Answers 3

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Answer 1 or 2 will do but the following should be added. The multiple functions you list make up a Set of functions. You have found that only one set of functions make a solution.

"The set of functions f(x) = 1, g(x) = 2x and h(x) = x + 2 is the only solution to the given problem."

It is common to use and in place of the last comma separating an enumerated list. You may even use both as in "g(x) = 2x, and h(x) = x + 2" according to Warner's English Grammar.

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  • I disagree. I could accept f(x) = 1, g(x) = 2x ,/and h(x) = x + 2 is the only solution to the given problem. Or your 'The set of ...'. But 1 and 2 sound outlandish.
    – Edwin Ashworth
    Commented Apr 13, 2021 at 15:40
  • Sorry, I added a comma to this answer by mistake (and then removed it).
    – DjinTonic
    Commented Sep 10, 2021 at 20:09
  • @DjinTonic; You and Oscar Wilde. Maybe switch to Absinthe.
    – Elliot
    Commented Sep 10, 2021 at 23:59
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For me, it sounds more natural to say x = 10, y = 20, and z = 15 is the/a solution when asked to find, say, three values in a problem. Therefore:

(Functions) f(x) = 1, g(x) = 2x, and h(x) = x + 2 is the only solution to the given problem.

(assuming that together they constitute one solution). "Functions" can be omitted if you like.

Or you could say

The system of functions f(x) = 1, g(x) = 2x, and h(x) = x + 2 is the only solution.

(Just as we refer to a system of equations).

Thus selecting arbitrarily a partial system of functions of the observations... we can ... ref

Clearly, if each function in a given set of generators belongs to a closed system F, then the system of functions generated by those generators is contained in F. Emil Post; The Two-Values Iterative System of Mathematical Logic p.14

Let ... be a system of functions defined on some measurable subset ... ref.

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The functions f(x) = 1, g(x) = 2x and h(x) = x + 2 are the only solutions to the given problem.

This is the only correct version.

"f(x) = 1, g(x) = 2x and h(x) = x + 2" stand in apposition to "functions". As such they do not influence the plurality of the subject which is "functions."

Functions is plural therefore the verb is "are".

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  • Would you say 'x = 2, y = 3 and z = 4 are the only solution to the problem'? I'll not add to the downvote, but the only sensible answer is to rewrite.
    – Edwin Ashworth
    Commented Oct 11, 2021 at 11:44
  • @EdwinAshworth the mathematics have clouded the issue for you. It might help if you omit them Would I say 'x = 2, y = 3 and z = 4 are the only solution to the problem'? Why do you ask, as I never said that? You yourself would say "These are the only solutions to the problem'? There is no need to rewrite.
    – Greybeard
    Commented Oct 11, 2021 at 17:34
  • No I wouldn't. The solution, the single solution, is {f(x) = 1, g(x) = 2x and h(x) = x + 2}. Here the solution set comprises three defined functions. A more typical solution set consisting of numbers is: the single solution to the simultaneous linear equations x + y = 4, x - 2y = 1 is {x = 3, y = 1}. I've still not added a downvote, though I see a second person has.
    – Edwin Ashworth
    Commented Oct 11, 2021 at 18:48

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