Composite functions

Question

Is this composite function legal?

g(x) : P({1,2,3,4})→{0,1,2,3,4} defined by g(X)=|X|

g(g(x))

My hunch is that it would not work since the image is a number and not a set.

Answer 1

By writing P({1,2,3,4}) do you mean the power set of the set {1,2,3,4}? If yes, then the composition of g with itself is not defined.