Nested Set subscript notation Suppose I have a Set $A$ with elements $P$:
$A = \{P_1,P_2,...,P_n\}$
And another Set $B$, where the elements are Set $A$:
$B = \{A_1,A_2,...,A_i\}$
How would I refer to a specific element $P$, as nested in $B$?
Would this be correct:
$P_{B{_i}n}$ ?
Thanks!
 A: I'm not quite sure what you are asking, but here's an attempt at an answer.  If you have a set $B$ with $i$ elements $A_1,\ldots,A_i$, and for each $j$ with $1 \le j \le i$ the set $A_j$ in turn has $n$ elements, we could denote these elements of $A_j$ by $P_{j,1},\ldots,P_{j,n}$.  So $B = \{A_1,\ldots,A_i\}$ and for each $j$ with $1 \le j \le i$ we have $A_j = \{P_{j,1}\ldots,P_{j,n}\}$.
It would then also be correct to write 
$$B = \left\{ \{P_{1,1},\ldots,P_{1,n}\},\ldots, \{P_{i,1},\ldots,P_{i,n} \} \right\}.$$
  This is similar to what you wrote, and it addresses the issue that you seemed to be assuming some unstated relation between a set denoted by $A$ and sets denoted by $A$ with various subscripts.
EDIT: As Henning says in his answer, the set $B$ itself does not contain any information about the order of its elements $A_1,\ldots,A_i$, and the set $A_j$ itself does not contain any information about the order of its elements $P_{j,1}\ldots,P_{j,n}$.  If you want an object that records this information, you could use the $i \times n$ matrix $(P_{j,k} : 1 \le j \le i \mathbin{\And} 1\le k \le n)$.  The sets $A_j$ are rows of this matrix, and the sets $P_{j,k}$ are entries of this matrix.
A: It looks like you're trying to refer to an element by its position in a particular set.
Unfortunately that is not mathematically meaningful -- the idea of a "set" does not include any concept of the elements of the set being arranged in any particular order, so speaking of, say, "the seventh element of the set" is not in itself meaningful.
If you want to speak of an object that consists of some element and a particular order these elements are considered in, you're free to do so -- just don't call what you have a "set", because that word means something else. More usual terms here would be a "sequence" or even a "list".
The notation $A=\{P_1,P_2,\ldots,P_n\}$ can be a bit confusing. What it does is not to define that $P_i$ is now notation for the $i$th element of $A$ (because there is no such thing). Rather, it says (in some contexts) two things at the same time


*

*"Assume that there are some things that we call $P_1$ through $P_n$".

*"Let's call the set whose members are exactly those things $A$".
This way $P_{23}$ is a name of something in its own right -- it is not related to the fact that this something happens to be an element of $A$, except incidentally by the fact that these two facts were introduced in the same breath.
