Trig polynomials countable?

In summary: 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.
  • #1
2,552
3

Homework Statement


Prove that the set of all trigonometric polynomials with integer coefficients is countable.

Homework Equations


[itex] t(x)= a+\sum a_ncos(nx)+ \sum b_n sin(nx) [/itex]
the sum is over n and is from 1 to some natural number.

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.
let's 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.
 
Physics news on Phys.org
  • #2
cragar said:

Homework Statement


Prove that the set of all trigonometric polynomials with integer coefficients is countable.

Homework Equations


[itex] t(x)= a+\sum a_ncos(nx)+ \sum b_n sin(nx) [/itex]
the sum is over n and is from 1 to some natural number.

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.
let's 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.

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
Dick said:
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.

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.
 

1. What are trig polynomials?

Trig polynomials are mathematical expressions that involve both trigonometric functions (such as sine, cosine, and tangent) and polynomials (such as x^2 + 3x + 2).

2. How are trig polynomials different from regular polynomials?

Trig polynomials involve trigonometric functions, while regular polynomials do not. This means that trig polynomials have periodic behavior and can be used to model cyclical phenomena.

3. Are trig polynomials countable?

Yes, trig polynomials are countable. This means that there exists a one-to-one correspondence between the set of trig polynomials and the set of natural numbers.

4. How is the countability of trig polynomials proven?

The countability of trig polynomials is proven by demonstrating that every trig polynomial can be uniquely represented by a finite sequence of natural numbers. This shows that the set of trig polynomials is the same size as the set of natural numbers, making it countable.

5. What is the significance of trig polynomials being countable?

The countability of trig polynomials allows us to easily categorize and analyze them, making them useful in various fields of mathematics and science. It also helps us understand the relationship between trigonometric functions and polynomials, leading to further discoveries and applications.

Suggested for: Trig polynomials countable?

Back
Top