Proper use of articles in mathematical expressions

Question

I am having trouble using articles correctly, especially in mathematical expressions.

I made two sentences:

1. Consider the family C of subsets of a set X.
2. Consider a family D of subsets of a set X.

It seems to me that 1) is saying that C contains all subsets of X, namely the power set of X, because a family that contains all of them is unique, while in 2) D need not contain all subsets of X (D can be the empty set). Is this reasonable?

Secondly,

1. Consider a function f defined by f(x) = x.
2. Consider the function f defined by f(x) = x.

I think only one of them is correct. The latter seems more adequate.

Thank you.

Answer 1

Do not say

Consider the family C of subsets of a set X

by itself, unless X has only one family of subsets. But you can say it if you go on to specify a unique one of the families:

Consider the family C of subsets of a set X defined by C = ...

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I would say

Consider the function f defined by f(x) = x

if there is only one such function.

I would not say

Consider a function f defined by f(x) = x

unless there is more than one such function; or at least we do not yet know whether there can be more than one such function.

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English has some peculiarities, though. We say

The set S has a least upper bound

and not

The set S has the least upper bound

even if we have proved S cannot have more than one least upper bound.

Answer 2

My own field is logic rather than mathematics, kin though they are.

In the first case, the use definite or indefinite article tells us only that "the" family C has already been mentioned or "a" family C is now being introduced. Either way sentence implies nothing as to whether there could be one family or more than one. What would make the difference would be to use "the subsets", which could be understood to mean the subsets. In that case, any other Family, X, with the subsets will be logically (and mathematically) identical with Family C. Otherwise, the use of the definite article before Family does not exclude the possibility of other families, E, F, G ... with different subsets of the same Class. The strict logical way of stating an identity is to say

Consider the family, C, such that (i) C contains the subsets of X and (ii) for any family C prime, if 'C prime' contains the subsets of X, then C prime is identical with C.

But the use of the definite article in the right place will do the job.

The second case is simpler, is a point where the languages of mathematics and logic converge. If the/a function f is "defined" by f(x)=x, then any function f prime, defined by f(x), will be identical with f. The use of the definite or the indefinite article can make no difference.