axiomatic sets 9 (power set)

The definition of a subset has be explained in the preceding post. Here, we shall use the definition.

If a set have many elements, there will be many subsets of the set.
Given a set $A$  which is not empty, a set whose elements are all subsets of $A$ is called the power set of $A$.
It is a set and expressed $\mathcal{A}$ .

\[ \forall x\exists z \forall w[w\in z\leftrightarrow w\subset x]   \]

It is axiom of power set. If you do not want to use the symbol $\subset$ ,
\[ \forall x\exists z \forall w[w\in z\leftrightarrow \forall y[y\in w\rightarrow y\in x ]] .    \]

If $A=\left\{ x,y  \right\}$ , then all subsets of $a$ is $\left\{ x  \right\}, \left\{ y  \right\},$  and  $\left\{ x,y  \right\}$ .
Therefore, $\mathcal{A}=\left\{ \phi, \left\{ x  \right\}, \left\{ y  \right\}, \left\{ x,y  \right\}  \right\}$ .

As you see, if the number of elements of a set is $n$ , the number of the elements of the power set is $2^n$ .
(including empty set $\phi$.)

As $\phi=0$ , the power set of $\left\{ 0,1  \right\}$ is $\left\{ 0,\left\{ 0,1  \right\}  \right\}=\left\{ 0, \left\{ 1  \right\}   \right\}$ .

The power set of $\phi$  is $\left\{ 0  \right\}$ ,
and the power set of the power set of $\phi$  is $\left\{ 0, \left\{ 0  \right\}  \right\}=\left\{ 0, 1  \right\}$ .