Can Turing Machines Enumerate Union and Cartesian Product of Constructible Sets?

[mathmari]

Hey!

We have that the sets $S_1$ and $S_2$ are constructible enumerable, that means that there is Turing machine that enumerates them, right?

To show that the set $S_1 \cup S_2$ is also constructible enumerable, we construct a Tuiring machine that enumerates alternately one element of $S_1$ and one of $S_2$.
Is this correct?

How can we show that the cartesian product $S_1 \times S_2$ is also constructible enumerable?

 

[Evgeny.Makarov]

We have that the sets $S_1$ and $S_2$ are constructible enumerable, that means that there is Turing machine that enumerates them, right?

I am not familiar with this term. Did you mean "recursively enumerable"?

To show that the set $S_1 \cup S_2$ is also constructible enumerable, we construct a Tuiring machine that enumerates alternately one element of $S_1$ and one of $S_2$.
Is this correct?

Yes.

How can we show that the cartesian product $S_1 \times S_2$ is also constructible enumerable?

You can use a pairing function from the proof that $\Bbb N\times\Bbb N$ is countable. Or you can use the fact that r.e. sets are semidecidable, i.e., accepted by some TM.