How To Show That B is Semi-Decidable Given A I am preparing for my Computational Theory final and ran into this exact problem :

B={ x | there exists a prefix of x that is in A}.

Show that B is semi-decidable.  In other words, you need to describe an algorithm M that on input a string x semi-decides if some prefix of x is in A (“semi-decides” means that it is OK if the algorithm loops for strings with no prefix in A). You will assume that you have an algorithm MA that semi-decides A.

Any help would be greatly appreciated!
A language is semi-decidable if there is a Turing machine that halts for all strings in the language, and doesn't halt for strings not in the language.

You are given that $A$ is semi-decidable. Therefore there is a Turing machine $M$ that halts on an input $x$ iff $x \in A$. Your goal is to construct another Turing machine $M'$ that halts on an input $x$ iff one of the prefixes of $x$ is in $A$. In other words, $M'$ should halt on $x$ iff $M$ halts on one of the prefixes of $x$.

You take it from here.