I have the following assignment question, and I'm having trouble even getting started:
Consider the set of functions $\mathcal{F}=\{f,g\}$, with $f:\mathbb{R}^2\to\mathbb{R}$, and $g:\mathbb{R}\to\mathbb{R}$, defined by $f(x,y)=xy$ and $g(x)=x+1$. Let $A=cl_\mathcal{F}(B)$, with $B=\{0\}$. I.e., $A$ is the closure of $B$ under the operations in $\mathcal{F}$. Then clearly $A=\mathbb{N}$. How would you go about showing that the following three equations, uniquely determine a function $h$?
$h(f(x,y))=f(h(x),h(y))$,
$h(g(x))=h(x)+2$, and
$h(0)=0$.
What is $A$ good for? Is it necessary to show what we're being asked to show? And also, we're on the topic of primitive recursive functions and such, so I don't understand how this would fit into that topic anyway.