Assuming the definition of a partial recursive function from $\mathbb{N}$ to $\mathbb{N}$, what is a recursive subset of $\mathbb{N}$ ? What is a recursively enumerable subset of $\mathbb{N}$ ?

Show that a subset $E \subseteq \mathbb{N}$ is recursive if and only if $E$ and $\mathbb{N} \backslash E$ are recursively enumerable.

Are the following subsets of $\mathbb{N}$ recursive?

(i) $\mathbb{K}:=\left\{n \mid n\right.$ codes a program and $f_{n, 1}(n)$ halts at some stage $\}$.

(ii) $\mathbb{K}_{100}:=\left\{n \mid n\right.$ codes a program and $f_{n, 1}(n)$ halts within 100 steps $\}$.