Give a recursive definition of:

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Homework Help Overview

The discussion revolves around defining recursive structures for specific sets, including odd positive integers, positive integer powers of 3, and polynomials with integer coefficients. The original poster presents initial recursive definitions for the first two sets but expresses uncertainty regarding the third set.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster provides recursive definitions for odd positive integers and powers of 3 but is unsure about how to approach the definition for polynomials with integer coefficients. Some participants suggest that ordering is essential for defining the recursive nature of polynomials.

Discussion Status

The discussion is ongoing, with participants exploring the requirements for a recursive definition, particularly focusing on the ordering of polynomials. There is no explicit consensus yet, but hints and guidance have been provided to help clarify the original poster's confusion.

Contextual Notes

Participants are grappling with the concept of recursion and the specific requirements for defining polynomials, indicating a need for clarity on the problem's expectations.

caseyd1981
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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?
 
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Welcome to PF!

caseyd1981 said:
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! :smile:

Hint: the first step is find a way of putting them in order. :wink:
 
Oh boy, I'm not sure that I follow...??
 
caseyd1981 said:
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? :smile:
 

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