The class of Turing$-$recognizable languages is closed under union. Let M$1$

and M$2$ be Turing Machines. Consider the following Turing Machine description:

M $=$On input w:

$($a$)$ Run M$1$ on input w$.$ If it accepts, accept.

$($b$)$ Run M$2$ on input w$.$ If it accepts, accept. Else reject.$$

As written, this construction cannot be used to argue that Turing$-$recognizable lan$-$

guages are closed under union. Give a brief explanation of what can go wrong with the

given construction, and how to modify the description of machine M so that it does

provide a valid argument that Turing$-$recognizable languages are closed under union