Why characteristic function is primitive recursive

Question

I'm studying recursive functions and right now I stucked in this:

"Natural numbers subset is PR if and only if characteristic function is PR". Why is that? Becouse it has values 0 ant s(0) only? So how about set {1, 2, 5} is it PR? If so, that means that all subsets are PR?

Thanks to reply to stupid question.

Answer 1

What you cited was the definition of PR subsets of $\mathbb N$. What it means is that in order to see if a set $S\subset\mathbb N$ is PR, you must see if it's characteristic function is PR.

There is no why here, since this is the definition.

The set $\{1,2,5\}$ is, by this definition, primitive recursive if and only if the function $$f(n) = \begin{cases}1 & \text{if } x=1,2,5\\0&\text{otherwise}\end{cases}$$ is primitive recursive.