# Referencing (multiple) objects within a collection of objects

In mathematical writing, when talking about multiple objects within a collection of objects, is it correct to say, "This set contains all objects x from the collection C such that x satisfies ."?

Specifically, is the expression "contains all objects x...such that x satisfies " correct? X is treated as singular here.

• Anything can be considered as set in naive Set Theory. So, yes, you can go ahead. – Ubi hatt Apr 3 '19 at 5:51
• @Ubihatt, I clarified the question; is it correct to treat 'x' as singular in the sentence? – Ar1Bis Apr 3 '19 at 7:36

Mathematician here. Your writing is correct, and I'm hoping I can lend some insight into why.

Here's how a mathematician may define S to be the set of all even integers. (You should know that "integer" means "a whole number, be it positive or negative or zero"; the letter Z is used to mean "the set of all integers"; and "x in Z" means "x is in the set of integers", as in "x is an integer".)

S = { x in Z | there exists a in Z with the property x=2a }

Aloud or in writing, I would say: "S is the set of all integers x such that there exists an integer a with the property that x=2a". That is, S is the set of all whole numbers that are precisely twice some whole number.

Okay, now why am I saying all this. Well, my main point is that x is a "dummy variable". Outside of the line of symbols above, x has no meaning. If I were to list out the elements of that set S, I would write:

S = { ... -6,-4,-2,0,2,4,6, ... }

Notice that I don't mention x anymore. It's irrelevant. The role that x plays is to stand in as an arbitrary and fixed object to describe what I want to include in the set S. If somebody comes along with some number, I could test whether it belongs in my set S by "plugging it in" for x and seeing whether it has the right properties:

• For example, if x=10: I see that 10 is in Z, and I see that 10=2*5 and 5 is in Z. Thus, 10 belongs in S.

• For another example, if x=11: I see that 11 is in Z, yet I see that 11 is NOT twice a whole number, so 11 does not belong in S.

• For a final example, if x=11.7: I see that 11.7 is not even in Z to begin with, so 11.7 does not belong in S.

In other words, the definition of S is based on x being some singular but unspecified object (that's the "arbitrary and fixed" idea), and we are declaring what properties that object must have to be included in the set.

• { x in C | P(x) is true }
• the set of all objects x in the set C that have property P(x)
• the set of all objects x in the set C that make P(x) true
• the set of all objects x in the set C such that P(x) holds true (or you could even just say "such that P(x)", with the "holds true" implied)
• the set of all objects x in the set C such that x satisfies property P

That is: yes, x is treated as singular, and I hope the above explanation (specifically the "arbitrary and fixed" notion) illustrates why.

• Thanks! Your explanation is very useful, especially the list of equivalent ways of expressing the same idea. – Ar1Bis Apr 5 '19 at 19:39

In your sentence there are two clauses.

In each clause there is the same verb form (V+s). This is the Present Simple Active (3rd person singular).

It means that the subjects (set and X) in both clauses are in the singular form.