State whether each of the following functions is injective, surjective, bijective, or none of these categories: (N is the set of natural numbers, and is the set of all strings over .)

(a) f : N N , f(n) = n div 3

(b) g : N N , g(n) = n 2 + 2n + 1

(c) h : , where = {a, b}, and h(w) = a |w| .

(d) f : N N , f(n) = n 1, if n is odd

n + 1 otherwise.

(e) h : , where = {a, b}, and h(w) = w R