## quarkine Group Title While studying the countability of sets, i came across the following problem (In Methods of Real Analysis, Goldberg) : Show that Pn=set of polynomials of degree n (with all coefficients being integers and n fixed positive integer) is countable.. 11 months ago 11 months ago

1. quarkine

my try was

2. Alchemista

All you need is that $$\mathbb{Z}^n$$ is countable because from there you can simply setup a bijection between $$\mathbb{Z}^n$$ and $$P_n$$ with integer coefficients .

3. Alchemista

This is indeed by induction because $$Z \times Z$$ is countable via cantor's pairing function. The same technique used to show that $$\mathbb{Q}$$ is countable. Then do the same thing inductively $$\mathbb{Z}^2 \times Z$$, etc.

4. Alchemista

Sorry I meant $$\mathbb{Z}^{n + 1}$$

5. quarkine

Hmm i guess that's true but i just needed to know if the method i mentioned above will work.I found it wont :(

6. quarkine