We want to count the number of elements in a set.
The number (or size) of a finite set is the unique natural number equivalent to it.
We denote the number of a set by $\num{A}$. Equally good notation, which we will not use in these sheets, is $\#(A)$.
If we restrict $E \mapsto \num{E}$ to the domain $\powerset{X}$ of some set $X$ then $\num{\cdot }: \powerset{X} \to \omega $ is a function.2