$L_0 = \{ \langle M, w, 0 \rangle \mid \text{$M$ halts on $w$}\}$
$L_1 = \{ \langle M, w, 1 \rangle \mid \text{$M$ does not halt on $w$}\}$$L = L_0 \cup L_1$
I need to determine where in the hierarchy of languages (recursive, recursively enumerable, not recursively enumerable) $L$ and its complement $\overline L$ belong. I reasoned as follows
$L = \{ \langle M, w, x\rangle \mid \text{$M$ halts on $w$ when $x=0$, $M$ doesn't halt on $w$ when $x = 1$, $x \in \{0, 1\}$}\}$
$L$ is clearly not recursively enumerable as a Turing machine wouldn't be able accept in all cases. It can accept only in case the input refers to $L_0$, but can't in case the input refers to $L_1$.
$\overline L = \overline L_0 \cap \overline L_1 = \emptyset$
Thus $\overline L$ is recursive.
Is my reasoning ok? This is a question from a previous exam paper.