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When stating the existence of universal objects in mathematics, one often has to write something like:

For every object X there exist an object Y and a map f : Y -> X such that [...] holds.

Here, both Y and the map f : Y -> X are part of the data so to speak. The object Y would be useless without the map f.

Question: Is the usage of there exist correct, or should it be there exists? I very often see the second one in textbooks, but I think the first one is correct. Does it change anything if I use a comma before 'and'?

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    If Y and f:Y -> X are different objects, then they are plural. – TimLymington May 16 at 8:11
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    Hello, abenthy. This really asks about whether subjects of the form A + B ('A and B') should ever be given singular verb agreement, and has been covered here before in depth. The search engine doesn't seem to work here as it used to, so I can't find the example I'm thinking of. However, 'Bacon and eggs is my favourite meal' and 'Health and safety is our prime concern' (for example) are both commonly agreed to have unitary (essentially single-concept, albeit with two components) subjects, and so to favour/require a singular verb. I'd say your example could equally well take either agreement. – Edwin Ashworth May 16 at 8:24
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    Note the conjunction and in "... an object Y and a map f ...." Naturally the verb is required to be in the plural: exist. HTH. This has nothing to do with math. – Kris May 16 at 8:25
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    It's a bit contentious. The structure is similar to there is/there are. 'There are an object and a map' sounds strange to some people's ears (mine included) because the first word is singular. So you'll often see that construction written in the singular. See: there is/ there are with a series of items. – S Conroy May 16 at 11:20
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    @SConroy Very helpful link, thank you – abenthy May 16 at 13:28
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You want to know whether in this case the conjunction "and" identifies a singular object composed of two parts, or two objects which are therefore plural.

As a hint:

  • if you can replace the word "and" with "with", "having" or a similar conjunction, without changing the sense, then it's singular.
  • If you can prefix the noun phrase with "both", "either", "all" "any" then it's plural

Example: Bacon and Eggs

  • both Bacon and Eggs: Plural
    • Both bacon and eggs are off the menu

There is no bacon, and there are no eggs either.

  • Bacon with Eggs: singular
    • Bacon with Eggs is an excellent meal
    • Bacon with Eggs is off the menu

Here there is potentially a different situation. Perhaps eggs are available on their own.

So "Bacon and Eggs" can be either singular or plural, but the meaning is not the same.

Example: there exist an object Y and a map ...

  • ... there exists an object Y having a map ...
  • ... there exist both an object Y and a map ...

Here, the situation is the same. The same objects exist, with the same relation. It's meaningless to consider the map in the absence of Y since Y is part of the map, so neither version would be unclear. It's up to you which you use.

However if it sounds awkward, you can improve it by substituting "having" for "and".

  • Thank you. So as I understand it, it is in this case accetable to write "there exists an object Y and a map f : Y -> X" as most people do, because it is clear from the context here that Y is part of the map. If you don't mind, I have one more question: Would you say a comma before 'and' makes the situation more clear here? – abenthy May 16 at 10:07
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    @abenthy If that's what most people do, then it's idiomatic, and you should do it too, so as to be better understood. Do most people use a comma? If so, then you should, but if not, then not. :-) – Ben May 16 at 10:37

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