Referencing (multiple) objects within a collection of objects

Solution 1:

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.

To address your question more directly, here are a few equivalent ways of writing/saying the idea you're asking about:

• { 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.