Why is a set not a partition of itself? Take the set {1,2,3,4}, why is {1,2,3,4} not a partition of this, which condition does it not meet?
By my understanding, a partition of a finite set $S$ is any set $\{ S_{1},...S_{n} \}$ of n subsets of $S$, which satisfy,


*

*$S_{i} \ne \emptyset$ for all $1 \leq i \leq n$,

*$S_{i} \cap S_{j} = \emptyset$ for all $1 \leq i,j \leq n$, $i \neq j$

*$S_{1} \cup \cdots \cup S_{n} = S$
Seeing as $S$ is a subset of $S$, which part of the definition breaks down here?
Note I think $\{\{1,2,3,4\}\}$ is a partition of $\{1,2,3,4\}$... could this be explained to?
 A: 
a partition of a finite set $S$ is any set $\{S_1, \dots, S_n\}$ of $n$ subsets of $S$...

You answered your own question. Which element of $\{1,2,3,4\}$ is a subset of $\{1,2,3,4\}$?
A: A partition of a set $A$ is a subset of its power set.
$\{\{1,2,3,4\}\}$ is, but $\{1,2,3,4\}$ is not, a partition of $\{1,2,3,4\}$.
$\{1,2,3,4\}$ is an element, not a subset of the power set of  $\{1,2,3,4\}$.
A: The following remarks do not aim at proving as a general law that no set is a partition of itself, but at showing via couterexamples that it is not the case that any set is a partition of itself, 
(1) A partition of a set S is, by definition, a family of sets. 
So, if a set S is not itself a family of sets, it cannot be a partition of any set; and consequently, it cannot be a partition of itself. 
(2) Lets consider a given set S that is a family of sets, say 
S = { {1, 2} , {2,3} }
Now suppose that S is a partition of itself. 
That would mean that  : Intersection(S) is empty 
( for by definition, the intersection of the elements of a partition is empty). 
But that is not true, for here we have 
Intersection (S) = {1,2} Inter {2,3} = {2} 
(3) If S = { {1, 2} , {2,3} } were a partition of itself, then Union (S) would be equal to S ( this  is also  a necessary condition to be a partition). 
But Union (S) = {1, 2} Union {2,3} = { 1,2,3} 
and this is not equal to S. 
