# Provide a recursive definition of the following sets

• Mar 2nd 2010, 10:41 AM
Runty
Provide a recursive definition of the following sets
I have some definitions that would apply to these, but I'm not sure whether or not they are very concrete, and I would also like to confirm that they are correct. Here are the sets and my current answers.

a) The set of odd positive numbers.
$\displaystyle 1\in S$
If $\displaystyle x\in S$, then $\displaystyle x+2\in S$

b) The positive integer powers of 3.
$\displaystyle 3\in S$
If $\displaystyle x\in S$, then $\displaystyle 3\times x\in S$

c) The positive multiples of 4.
$\displaystyle 4\in S$
If $\displaystyle x\in S, y\in S$, then $\displaystyle x+y\in S$

Are these all accurate, or would they benefit from more detailed answers?
• Mar 2nd 2010, 03:12 PM
MollyMillions
You've got the right idea, but it might be useful to use a bit different notation, since a recursion is asked for. Something like this for c) maybe:

Let $\displaystyle S$ be the set of positive multiples of 4.
$\displaystyle s_n \in S$
$\displaystyle s_0 = 4$
$\displaystyle s_n = s_{n-1} + 4$