# Homework Help: Trig polynomials countable?

1. Sep 13, 2013

### cragar

1. The problem statement, all variables and given/known data
Prove that the set of all trigonometric polynomials with integer coefficients is countable.
2. Relevant equations
$t(x)= a+\sum a_ncos(nx)+ \sum b_n sin(nx)$
the sum is over n and is from 1 to some natural number.
3. The attempt at a solution
So basically we have to look at all the possible trig polynomials of all finite lengths.
with some natural a out front.
lets first look at the ones where n=1
so we have a+cos(1x)+bsin(x)
How about we map these to the first prime number 2.
since a can be anything, if a is 1 then this t(x) will go to 2 and if a is 2 then t(x) will go to 2^2
if a is three then t(x) goes to 2^3.
Now for the sum from n=1 to 2 will map these to the next prime and do the same process as above with the constant out front. Is this the right idea.
I am mapping these to prime numbers so we can get a unique mapping and we don't have to worry about sending 2 things to one thing.

2. Sep 14, 2013

### Dick

In general the set of all finite subsets of a countable set is countable. That's the proof you want. There's nothing about trig polynomials that makes it any easier. Try to prove that first.

3. Sep 14, 2013

### Zondrina

Going from the hint Dick has given, start by showing that the set of subsets with no more than $n$ elements is countable. Since you're working finite here, induction should be good I believe.