Re: Sets question (maybe)
- From: thomasjack <thomasjack@xxxxxxxxx>
- Date: Tue, 23 Oct 2007 16:15:02 -0000
On Oct 23, 9:22 am, Name And Address Supplied
<name_and_address_suppl...@xxxxxxxxxxx> wrote:
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
This is just a notational problem. The way I've seen it is: let J and
I be sets. J is indexed by I if there is given a surjective function
f:A->B. For i in I, denote f(i) by j_i (j sub i) and denote the
indexing of J by I as {j_i}_(i in I) (...this last is written without
the parentheses and with the little symbol that looks like an epsilon
instead of "in").
Then you can just treat J as a set and do cartesian products and
whatever else with it, and you can also pick out the ith element by
referring to j_i.
.
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