Give a recursive definition of:

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In summary, a recursive definition of the set of odd positive integers is f(0)=1, f(n)=f(n-1)+2 for n>=1. A recursive definition of the set of positive integer powers of 3 is f(0)=1, f(n)=3f(n-1) for n>=1. For the set of polynomials with integer coefficients, a possible recursive definition would be to put them in order by degree, starting with constant polynomials and then adding a new term with each recursive step. For example, the first polynomial would be 0, the next would be 1, the next would be x, and so on, with each new polynomial adding a term with a higher degree than the
  • #1
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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  • #2
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:
 
  • #3
Oh boy, I'm not sure that I follow...??
 
  • #4
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:
 

1. What is a recursive definition?

A recursive definition is a definition that is based on itself. It is a way of defining a concept or mathematical object by using the same concept within its own definition. This creates a self-referencing relationship between the concept and its definition.

2. How is a recursive definition different from a non-recursive definition?

A non-recursive definition is a straightforward definition that does not refer to itself. It defines a concept or object in a direct and simple way. On the other hand, a recursive definition uses the concept or object being defined within its own definition, making it more complex but often more powerful.

3. Can you give an example of a recursive definition?

Yes, the Fibonacci sequence is a classic example of a recursive definition. It is defined as: F(n) = F(n-1) + F(n-2), where F(0) = 0 and F(1) = 1. This means that each number in the sequence is defined by adding the two previous numbers, starting with 0 and 1.

4. What are the advantages of using a recursive definition?

Recursive definitions can often be more concise and elegant than non-recursive definitions. They also allow for the creation of complex structures and patterns that may not be possible with non-recursive definitions. Additionally, recursive definitions can make it easier to understand and solve certain problems or equations.

5. Are there any drawbacks to using a recursive definition?

While recursive definitions can be powerful, they can also be more difficult to understand and analyze. In some cases, they may also lead to infinite loops or excessive computation, which can be inefficient. It is important to carefully consider if a recursive definition is the best approach for a given problem or concept.

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