Give a recursive definition of the set of polynomials with integer coefficients.
What I don't understand here is what is the set of polynomials with integer coefficients? Can someone give me an example?
I believe the following would be a recursive definition of the set of polynomials with integer coefficients:
$\displaystyle 0 \in S $(some arbitrary set)
If $\displaystyle p(x) \in S $, then $\displaystyle p(x) + cx^n \in S $, $\displaystyle c \in \mathbb{Z}, \ n \in \mathbb{Z}, \ n \geq 0 $.
The set of polynomials with integer coefficients is for example $\displaystyle \{x, x+1, x^2,2x^2+6x+7, x^3, \ldots \} $.