Can the image of a cardinality function be a set?

Question

Let a function f(x)=|x|, where the domain is that x is an element of the power set of {2, 3, 4, 5}. Since any input into the function could result in a cardinality from 1-4, could you say that the range/image of function f(x) is {1, 2, 3, 4} since the only possible cardinalities are from 1-4?

Answer 1

Let $f:\mathcal{P}(\{2,3,4,5\})\rightarrow \mathbb{N},\ f(x)=|x|.$

Notice that $\color{red}{\emptyset}\in\mathcal{P}(\{2,3,4,5\})$, so you have to consider also $f(\color{red}{\emptyset})=|\color{red}{\emptyset}|=0 \implies 0 \in \mathrm{Im}(f).$

Hence, $\mathrm{Im}(f)=\{0,1,2,3,4\}.$