In each case below, determine whether the given set is countable or uncountable. Prove your answer.
a. The set of all real numbers that are roots of integer polynomials; in other words, the set of real numbers x such that, for some nonnegative integer n and some integers a0, a1, ... , an, x is a solution to the equation
b. The set of all non decreasing functions from N to N.
c. The set of all functions from N to N whose range is finite.
d. The set of all non decreasing functions from N to N whose range is finite (i.e, all “step” functions).
e. The set of all periodic functions from N to N . (A function f : N →
N is periodic if, for some positive integer Pf , f (x + Pf ) = f (x) for every x.)
f. The set of all eventually constant functions from N to N . (A function
f : N → N is eventually constant if, for some C and for some N, f (x) = C for every x ≥ N.)