Is this set finite, countably infinite, or uncountable:
a) $V = \{f: \mathbb{N} \to \mathbb{N}\}$ (the set of all functions that map each natural number to a natural number.
b) $\mathbb{Z}^3=\{(a,b,c):a,b,c∈\mathbb{Z}\}$ (the set of triples of integers)
Can I just say that we can create a mapping for each triplet in the sense that $(0,0,0) \to 0$, $(1,0,0) \to 1$, $(1,0,1) \to 2$ etc... every triplet will be accounted for and only once (bijection). Therefore, it is countable?
c) $T = \{p(x) : p(x) = a_nx^n +\cdots + a_1x + a_0\mid n \in \mathbb{N},\; a_0, a_1, \ldots, a_n \in \mathbb{Z}\}$ (the set of all polynomials with integer coefficients, of any degree)
I believe you use induction on just a polynomial of degree 2 (whose countability proof is identical to the question above)...but am not exactly getting it
Help appreciated please.