I have two infinite languages, A and B, and they're disjoint.
A is not Turing recognizable, and B is decidable.
What's the result of their union? meaning, is it a decidable/recognizable/not recognizable language?
I tried thinking of some examples and I haven't gotten to a decisive answer.
The union is not Turing recognizable. A simple reduction from $A$ to $A\cup B$ operates as follows. Given input $x$, the reduction checks if $x$ is in $B$ ( this is possible as $B$ is decidable) and outputs $x$ if $x\notin L(B)$; Otherwise, if $x\in B$, it outputs some fixed $y$ that is not in $A\cup B$.
First, note that $y$ exists as $A\neq \overline{B}$. Indeed, otherwise, $A$ would be decidable. Then, it is not hard to see that if $x\in L(A)$, then as $A$ and $B$ are disjoint, we get that $x\notin B$, and so the reduction outputs $x$ which is in $ A\subseteq A\cup B$. Conversely, if $x\notin L(A)$, then we always output a word not in $A\cup B$. Indeed, we output the word $x$ if $x\notin B$, and we output $y$ otherwise. In both cases, the output is not in the union $A\cup B$.
