Why a unit set is not the same as its element? $\{x\} \ne x$? I was solving a list of exercises from homework and I answered one question wrong because I thought that a singleton set and its element were the same thing...
I found only a few results from a Google search:
https://www.google.com/#q=%22set+is+not+the+same+as+its+element%22
May anyone please explain this to me?
 A: First of all, it can't always be the same. $\{\varnothing\}\neq\varnothing$. On the right-hand side, we have the empty set which has no elements but on the left-hand side we have a set which does include an element.
More generally, one of the axioms of modern set theory allows us to prove that no set is an element of itself. In particular $\{x\}\notin\{x\}$. Since the only element of $\{x\}$ is $x$, when we say $\{x\}\notin\{x\}$ we actually say that $x\neq\{x\}$.
A: If a singleton set and its element were the same thing, then the set $\{\{1,2,3\}\}$, which has just one member, would be the same thing as the set $\{1,2,3\}$, which has three members.  Suppose one wants to enumerate the partitions of the set $\{1,2,3\}$.  They are:
\begin{align}
& \{\{1,2,3\}\} \\[6pt]
& \{\{1,2\},\  \  \{3\}\} \\[6pt]
& \{\{1,3\},\  \  \{2\}\} \\[6pt]
& \{\{2,3\},\  \  \{1\}\} \\[6pt]
& \{\{1\},\  \  \{2\}, \  \  \{3\}\}
\end{align}
The first partition listed has one part, not three.  That part has three elements, not one.
A: A bag containing an orange is not the same thing as the orange itself.
