Mathematical Definition of a function with array domain and codomain

Question

I know the definition of a simple function is: $f:\biggl\{\begin{array}{rl}R&\to R\\ x&\mapsto f(x)\end{array} $

But i want to define a function which get a set as input and make the result be a set. $G:\biggl\{\begin{array}{rl}?&\to ?\\ \{x_1,....,x_?\}&\mapsto \{a_1,...,a_?\}\end{array} $//the size of input set and output and depend on the input

assume $x_i \in X$ and $a_i \in A$

could you please help me to define the domain and co domain?

Answer 1

$G$ maps subsets of $X$ to subsets of $A$ :

$$G:\mathcal{P}(X)\to\mathcal{P}(A)$$

If you have a function $g:X\to A$, you can define $G(Y)=\lbrace g(y),\;y\in Y\rbrace$