(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let P(n) be the set of all polynomial of degree n with integer coefficients. Prove that P(n) is countable, then show that all polynomials with integer coefficients is a countable set.

2. The attempt at a solution

For this problem the book gives me a hint that using induction is one way to prove this. So by going off this I say that P(0) is countable since it is the set of all constants. After this I say that P(1) is countable since P(1) = ax + P(0) in which a ε A and A = {z: z ε Z, z ≠ 0}. Now my problem is that I do not know how to make the jump from P(1) to P(n) and then to P(n+1). For the second part of the question I know that all polynomials with integer coefficients are countable since if we were to take a union of all the sets they would be countable since the union of countable sets are countable.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Proof of Polynomial Countability

**Physics Forums | Science Articles, Homework Help, Discussion**