# “There exist” vs “there exists” for universal objects in mathematics

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’?

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”.