Question: Let X = Map (N, R) (X is set of functions from Natural numbers to real numbers). The function c: X -> X is defined by (c(f))(x) = f(x^2). Prove that c is a surjection.
Proof:
Consider g ∈ X. Then g is a function from N to R. We have to show that there exists f ∈ X such that c(f) = g i.e. f(x^2)= g(x) for every natural number x.
Let us define f as follows. f: N -> R where f(0) = g(0) and f(1) = g(1).
So, f(r)={g($\sqrt{r}$); if r is a perfect square
{ 1; otherwise
Thus f is a pre-image of g under c.