Give a recursive definition of:

[caseyd1981]

Give a recursive definition of

a) the set of odd positive integers
b) the set of positive integer powers of 3
c) the set of polynomials with integer coefficients


I have the first two:
a) f(0)=1, f(n)=f(n-1)+2 for n>=1
b) f(0)=1, f(n)=3f(n-1) for n>=1

For c, I am not even quite sure exactly what it is asking or where to begin?

 

[tiny-tim]

Welcome to PF!

Give a recursive definition of
c) the set of polynomials with integer coefficients
…
For c, I am not even quite sure exactly what it is asking or where to begin?

Hi caseyd1981! Welcome to PF!

Hint: the first step is find a way of putting them in order.

 

[caseyd1981]

Oh boy, I'm not sure that I follow...??

 

[tiny-tim]

Oh boy, I'm not sure that I follow...??

If it's recursive, you must put them in order, so that you know which is the next one at each stage …

and you can't put them, for example, in the order x+1, x+2, x+3, … , going "up to infinity", and then start on 2x+1, 2x+2, 2x+3, …, because 2x+1 won't be the next one to anything.

So you need a way of putting them in order, without ever "going off to infinity" and leaving some behind for later.

How can you do that?