Partial recursive set from values of recursive function

Question

So I know that $E = \{n\ |\ \text{if } F(n,n) \text{ is defined}\}$, is recursively enumerable set which isn't recursive. But how i find a recursive function which values is that set?

Answer 1

Fix some $m$ such that $F(m,m)$ is defined. Then define a function $g$ by setting $$ g(n,s)=\begin{cases} n&\text{if the computation of }F(n,n)\text{ produces an output in }<s\text{ steps.}\\ m&\text{otherwise.} \end{cases} $$ Then $g$ is recursive, and its range is your $E$.

Incidentally, I don't recall ever hearing "partial recursive set". You probably meant "recursively enumerable set". Partial recursiveness is a property of functions, not of sets.