How does a disjoint union differ from a set of sets?

[pellman]

Given an indexed collection of sets A_x the disjoint union of these sets can be thought of as the ordinary union of the sets \{ x \} \times A_x for all x. That is, it is the set of all pairs (x, a) where a \in A_x.

(Correct me at this point if my understanding of disjoint union is wrong.)

Does this have any practical difference from set of all A_x ?

Denote the set of index values by X. That is, is there any practical difference between \{ (x, a) | x \in X \wedge a \in A_x \} versus \{ A_x | x \in X \} ?

 

[FactChecker]

Clearly the elements of {(x,a)|x∈X∧a∈A_x} are all 2-tuples and the elements of {A_x|x∈X} are all sets. And the first has many more elements. I guess the practicality of each depends on what is done with them. If two of the sets are identical (A₁=A₂), {A_x|x∈X} would not distinguish between them, whereas (1, a) ≠ (2, a).

 

[pellman]

They are certainly not identical in a set-theoretic sense. But they are equivalent, it seems to me, in that each set consists of all the pairs (a,x). In the set of sets case, if we choose an element of any A_x , we get the value of x associated with it by virtue of being an element of A_x. Both the disjoint union and the set of sets seem to me to contain the same information.

 

[FactChecker]

They are certainly not identical in a set-theoretic sense. But they are equivalent, it seems to me, in that each set consists of all the pairs (a,x). In the set of sets case, if we choose an element of any A_x , we get the value of x associated with it by virtue of being an element of A_x. Both the disjoint union and the set of sets seem to me to contain the same information.

Not if you want to know how many times the same set is repeated. A set is a collection where repitition and order are ignored. If A₁=A₂, you would say that {A₁, A₂} is just {A₁}. That would lose track of repeated sets. But {(x,a)|x∈X∧a∈Ax} would keep them distinct. So there is loss of information in the simple "set of sets" approach. Also if the index gives information about the order of the sets, that information would be lost.

 

[pellman]

Not if you want to know how many times the same set is repeated. A set is a collection where repitition and order are ignored. If A₁=A₂, you would say that {A₁, A₂} is just {A₁}. That would lose track of repeated sets. But {(x,a)|x∈X∧a∈Ax} would keep them distinct. So there is loss of information in the simple "set of sets" approach. Also if the index gives information about the order of the sets, that information would be lost.

For a set A_x we have the set itself, that is, the collection { a , b, c ,.. } whatever it is, and name of the set "A_x", which includes the index x. So you are saying that when we take the set of sets A_x, what we really end up with is { {a,b,c,..},...} and we lose the names.
That makes sense now. If we had A = {1,2}, B={3,4}, C={3,4}, then the set of these sets is {{1,2},{3,4}} not {A,B,C}.

Thank you for your reply.