(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.

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# Homework Help: Proof of Polynomial Countability

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