Constructible enumerable set We suppose that the sets $S_1$ and $S_2$ are constructible enumerable, that means that there is an algorithm that enumerates them. Show that the sets $S_1 \cup S_2$ and $S_1 \times S_2$ are also constructible enumerable. 

To show this do we have to construct a Turing machine for the union and one for the cartesian product?
Hints:


For $S_1 \cup S_2$, use the same idea in the proof that the natural numbers and the integers have the same cardinality. The proof goes by listing the integers in sequence:
$$ 0,1,-1,2,-2,3,-3,\ldots $$
You can think of $S_1$ as $0,1,2,3,\ldots$ and of $S_2$ as $-1,-2,-3,\ldots$.
For $S_1 \times S_2$, use the same idea in the proof that the natural numbers and the positive rationals have the same cardinality. The proof uses the "diagonal method", and consists of listing the rationals sorted by the maximum of the numerator and the denominator:
$$ \frac{1}{1}, \frac{1}{2}, \frac{2}{1}, \frac{2}{2}, \frac{1}{3}, \frac{2}{3}, \frac{3}{1}, \frac{3}{2}, \frac{3}{3}, \ldots $$
In this case $S_1=S_2$ are both $1,2,3,\ldots$.