Should set representation be used to show sample spaces? I have seen several books representing the sample spaces as sets :
eg. Sample space for outcomes of a fair die :

S = { 1,2,3,4,5,6 }

And then 
P(1) = 1 / |S|
P(2) = 1 / |S|
.
.
P(6) = 1 / |S|
This would work fine in cases all the outcomes are unique.
How about I have a die with 6 sides, only one of which is 1 and rest all 2.
Then sample space representation should be 

S = {1,2} 

or 

S = {1,2,2,2,2,2} ?

P(2) = 5 / |S| ?
But |S| = 2 in both cases (Counting only the unique values).
However I understand that sample space size should still be 6.
From what I know sets representation should show only the unique values.
Kind of confused, can someone help ? 
 A: Quick answer: In my opinion no, sample spaces should not be enumerated.
Long version: I prefer to view the sample space as an abstract set, e.g. for an idealized dice I could use $S = S_1 \cup S_2 \cup S_3 \cup S_4 \cup S_5 \cup S_6$ where $S_i \cap S_j = \emptyset$ for $i \neq j$
The elements of $S_1$ are not specified in mathematical terms, but for modelling the real world $S_1$ will model all events which lead to the side 1 to be on top (e.g. "The cube lies with side one up and side two to the west" or "The cube lies with side one up and side two to the north")
The assignments of values is then done by a random variable, i.e. a mapping form $S$ to the (real) numbers. In your first example this is 
$$X(s) = \left\lbrace\begin{array}{ll}
1& \text{if }s\in S_1\\
2& \text{if }s\in S_2\\
3& \text{if }s\in S_3\\
4& \text{if }s\in S_4\\
5& \text{if }s\in S_5\\
6& \text{if }s\in S_6\\
\end{array}\right.$$
your second example could use
$$Y(s) = \left\lbrace\begin{array}{ll}
1& \text{if }s\in S_1\\
1& \text{if }s\in S_2\\
1& \text{if }s\in S_3\\
1& \text{if }s\in S_4\\
1& \text{if }s\in S_5\\
2& \text{if }s\in S_6\\
\end{array}\right.$$
The probability is a measure on S, i.e. a particular mapping of subsets of S to numbers. 
The convenient notation $P(X=1)$ is short for the more rigurous $P(\{s \in S: X(s) = 1\})$, which reduces to $P(S_1)$. For a fair dice you assume $P(S_1)=P(S_2)=P(S_3)=P(S_4)=P(S_5)=P(S_6)=\frac{1}{6}$
It may look just like a different notation, but in my oppinion it is better to strictly distinguish between abstract elements of the sample space on one hand, and numbers on the other hand. The mapping between the two is formalized in the notion of a random variable.
