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Your exam question makes very little sense. The obvious reading would be this: Let $M$ and $N$ be two Turing machines. Why is it not possible to prove that $M$ and $N$ compute the same function? More precisely: It is not the case that for all Turing machines $M$ and $N$ it is provable that $M$ and $N$ compute the same function. Well, this is quite ...
If $A=\{0,1\}^*$ then $A\cup B=\{0,1\}^*$, regardless of what $B$ is. $\{0,1\}^*$ is decidable, and choosing to express it as something involving undecidable things doesn't change that fact. The question "Is $w$ in $A\cup B$?" is equivalent to "Is at least one of the following statements true? $w$ is in $A$; $w$ is in $B$; $w$ is in both $A$ and&...