Sets question (maybe)

Dear all,

I wonder if you can help with the following problem. I'm not a
mathematician, so please bear with me.

Informally speaking: I have a set of, say, 5 entities, and I wish to
give each a unique index j. In particular, I want to pick one entity
-- it doesn't matter which -- and I want to designate that j = 1. Then
I want to pick another, and index that one j = 2. And so on until I
have five individuals indexed 1 to 5. Now I want to order this list,
so that it runs from 1 through to 5.

Q1: What do I call this list? I think it is not a set in the strict
sense, as a set is not ordered, and also a set with repeated elements
is equal to the same set with the repeated elements removed. But I
want to describe the object J = {1, 2, 3, 4, 5}, and say things like
"j E J" (j is a member of J). I also want to say that the jth element
of J is j -- this of course would not be true if J wasn't ordered /
had repeated elements. So, what is J? A set, a list, a group, what?

Q2: I don't know that there are actually 5 entities, but I do know
there is a finite number of them. I want to index them as above, and I
want to say in a mathematically compact way that the indices are the
natural numbers running from 1 to the maximum index value (which is
equal to the number of entities I want to index). In particular, I
don't want to have to assign a symbol to denote the total number of
entities.

Q3: If J were a set, and K = {1,2} were some other set, I could define
a Cartesian product JxK = {{{1,1},{1,2}},{{2,1},{2,2}},{{3,1},{3,2}},
{{4,1},{4,2}},{{5,1},{5,2}}}. But if J and K are not sets, but are the
type of objects I've described above, can I still make this operation?
Or is there an analogous operation, and if so, what is it called?

Q4: Is the "curly bracket" notation okay for what I am doing, i.e. "{"
and "}"?

Many thanks in advance,

NAAS

.

Relevant Pages

  • Re: Sets question (maybe)
    ... mathematician, so please bear with me. ... is equal to the same set with the repeated elements removed. ... Or is there an analogous operation, and if so, what is it called? ...
    (sci.math)
  • Re: Sets question (maybe)
    ... mathematician, ... is equal to the same set with the repeated elements removed. ... Sounds like a sequence to me. ... If you really mean the jth element is j, then you have an initial segment of the naturals. ...
    (sci.math)
  • Re: Sets question (maybe)
    ... mathematician, ... is equal to the same set with the repeated elements ... In the case where f also happens to be injective, f is an enumeration *without* repetitions. ...
    (sci.math)