power sets

If the element of a set is also a set, the set whose elements are a set is called a power set.

Given a set $X=\left\{1, 2, 3 \right\}$ , the number of all subsets of $X$ is eight,
$\phi, \left\{1 \right\}, \left\{2 \right\}, \left\{3 \right\}, $
$ \left\{1,2 \right\}, \left\{2,3 \right\}, \left\{1,3 \right\}, \left\{1,2,3 \right\}$ .

Therefore, the power set of $X$  is the set whose elements are all above.
$\left\{\phi, \left\{1 \right\}, \left\{2 \right\}, \left\{3 \right\}, \left\{1,2 \right\}, \left\{2,3 \right\}, \left\{1,3 \right\}, \left\{1,2,3 \right\}  \right\}$ 

In general, if the number of the elements of a finite set is $n$ , the number of the elements of the power set becomes $2^n$ . It will be understood easily.

As this sample is very easy, you should visualize the power set of $[0,1]$
(the interval of real numbers). You will find it is impossible to imagine or write the result.
But it exists definitely.

If the set $X$  is the empty set $\phi$ , the power set of $X$  is $\left\{\phi \right\}$ .
Strictly, this power set of $X$  is not the empty set. That is to say,
$\left\{\phi \right\}\ne \phi$ .
Because the power set of $X$  has one element, but $X$  has no element.