Consider the following four rules that recursively define set $S.$

Rule $1$: $$ is an element of $S.$

Rule $2$: If $x$ is an element of $S,$ then $x$ is an element of $S.$

Rule $3$: If $y$ and $z$ are elements of $S,$ then $yz$ is an element of $S.$

Rule $4$: All of the elements of $S$ can be obtained using rules $1,2,$ and $3$ and repetitions of those rules.

Which of the following is an element of set $S?$

A$)$

B$)$

C$)\frac{}{{?}_{\frac{?}{?}}}\frac{}{{?}_{\frac{?}{?}}}A$

D$)$

E$)$