is the set of all polynomials of degree n with integer coefficients.
Prove that is countable.
By induction:
Since a is an integer, we can put the integers in a 1-1 correspondence with the naturals numbers. Namely, define
Assume P(k) is true for a fixed but arbitrary , where P(k) is defined as
Prove P(k+1) is true
I need help in order to proceed.
In the induction step, you need to show that the Cartesian product of two countable sets (the set of the new leading coefficients, which is , and the set of polynomials of the previous degree, which is countable by the induction hypothesis) is countable. See this thread. It contains the exact formula for the bijection, but you may not need it.
What is n here?Assume P(k) is true for a fixed but arbitrary
First, this is not a good definition because have not been defined before. Is P(k) a fixed polynomial or a set of polynomials? In either of those cases, P(k) cannot be true or false.