Prove the decidability of an enumerable set. A and B are enumerable, C is decidable.

Question

Sets A and B are enumerable, C is decidable. $A$ $\subseteq$ $C$ $\subseteq$ $A$ $\cup$ $B$. $A$ $\cap$ $B$ = $\emptyset $.Prove that A is decidable too

Answer 1

Here is an algorithm to determine if a number $n$ is in $A$ or not. First of all, ask if $n\in C$: if $n\not\in C$, then $n\not\in A$ since $A\subseteq C$. If instead $n\in C$, we proceed as follows: we ask at the same time if $n\in A$ and if $n\in B$. Notice that this question will be answered in a finite amount of time, since $C\subseteq A\cup B$, and exactly one of the two questions will have positive answer, since $A\cap B=\emptyset$. Hence, $A$ is decidable.