Recognizer for decidable language and words it doesn't halt on Suppose we have a decidable language B (there exists some TM that decides it). Suppose we have another TM M which only recognizes B but does not decide it - there are words not in B that M doesn't halt on (for words not in B M might either reject or not halt).

Let A = {w | M does not halt on the word w}.

Which of the following has to be true:

1) A must be decidable

2) A might not be decidable but it has to be recognizable

3)  A might not be recognizable but its complement has to be recognizable.

4) It might be that A and its complement are both not recognizable.

I was asked this question and i think that the answer is 3. 

What i don't understand is why give the extra information about some decidable language B that M recognizes if the complement of A will always be recognizable regardless of whether it recognizes some decidable language B -> for any TM, the set of all the words it does halt on will always be recognizable. The recognizer for A complement will be - run M on w and accept if it halts).

What am i missing ? Is the info about B redundant ?
The question is designed to test your understanding, so it's legitimate regardless of whether the details of&nbsp;$B$ are actually relevant. Maybe you're supposed to understand why they're relevant; maybe you're supposed to understand why they're not.

You've said that "recognize" is defined to mean that the Turing machine accepts all words in the language, and either rejects or loops on words not in the language.  This means that $A$&nbsp;is some subset of $\overline{B}$ (the complement of&nbsp;$B$). Since $B$&nbsp;is decidable, we know that&nbsp;$\overline{B}$ is also decidable.  So, $A$&nbsp;is a subset of some decidable language.  Well, $\Sigma^*$&nbsp;is a decidable language, so $A$&nbsp;could be any subset of&nbsp;$\Sigma^*$. So it looks like $A$&nbsp;could be any language at all! However, we have a little more information: $\overline{A} = \{w\mid M(w)\text{ halts}\}$, so $\overline{A}$&nbsp;is recognizable.