Recognizer for decidable language and words it doesn't halt on
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.