Just a definition-question; (about powerset and a set)

Question

idku · Answer

So, suppose I have a set $\color{black}{\small \rm\displaystyle B}$, and a power-set of this set $\color{black}{\small \rm\displaystyle B}$, denoted as $\color{black}{\small \rm \displaystyle P(B)}$. Suppose also that $\color{black}{ \displaystyle x}$ is an arbitrary element in $\color{black}{ \displaystyle B}$. Is it true then, that $-$  if $\color{black}{ \displaystyle x\in B}$ then $\color{black}{ \displaystyle x\in P(B)}$?

ganeshie8 · Answer

Short answer is no.

Do you see any difference between $x$ and $\{x\}$ ?

idku · Answer

I think that ${x}$ is an element, and $\{x\}$ is a set.

ganeshie8 · Answer

Yes, what's the definition of a powerset ?

idku · Answer

The set of all possible subsets of the set.

ganeshie8 · Answer

Notice that a subset is a set.

idku · Answer

Yes, I know, and that what I was asking $-$ whether or not $x$ can also be called a set or not.

idku · Answer

because, if not, the statement is false, and if yes, the statement is true.

ganeshie8 · Answer

$x$ and $\{x\}$ are two different things.

$x$ is an element,
$\{x\}$ is a set with one element $x$.

idku · Answer

So, $x$ is definitely $\color{red}{\bf not}$ a set?

idku · Answer

(all I want to verify)

ganeshie8 · Answer

Yes.

$x\not \in P(B)$

The elements of the powerset of B are subsets of B, not elements of B.
$\{x\} \in P(B)$

idku · Answer

I see, so if I make either of the two statements below, the they are false?

$*$        If $\color{black}{ \displaystyle x\in B}$, then $\color{black}{ \displaystyle x\in P(B)}$.
$*$        If $\color{black}{ \displaystyle x\in P(B)}$, then $\color{black}{ \displaystyle x\in B}$.

ganeshie8 · Answer

Both false

idku · Answer

Yes, thank you for confirming! (Now I am ready to go with my proof)
Have a nice weekend!

ganeshie8 · Answer

Np, you too have a nice weekend :)