Writing power set of $\{\emptyset,x,y,z\}$ That $\emptyset$ is confusing me, should we consider $\emptyset$ as an element or should we discard it?
So here is my work:
$\mathcal P(\{\emptyset,x,y,z\})=\{\emptyset,\{ \emptyset \},\{x  \} ,\{ y \},\{ z \},\{ \emptyset ,x \},\{ \emptyset, y \},\{ \emptyset,z \},\{ x,y \},\{ x,z \},\{ y,z \},\{\emptyset,x,y  \},\{ \emptyset,x, z \},\{\emptyset,y,z  \},\{ x,y,z \},\{\emptyset,x,y,z  \}    \}$

The power set of an $n$-element set always has $2^n$ elements. The empty set is also always an element in the power set of any set.
So yes, your work is correct, and since $\varnothing$ is an element of the original set, it gets considered like any other element.
