Let S1 and S2 be disjoint countably infinite sets. Show that S1 U S2 is countable.
Suppose that S1; S2; S3 .... are disjoint countably infinite sets. Show that U∞ & n=1 Sn is countable.
What if the sets above are not disjoint or infinite, are the conclusions still valid?

Question

Let S1 and S2 be disjoint countably infinite sets. Show that S1 U S2 is countable.
Suppose that S1; S2; S3 .... are disjoint countably infinite sets. Show that U∞ & n=1 Sn is countable.
What if the sets above are not disjoint or infinite, are the conclusions still valid?

kinggeorge · Answer

Well, you would need to show that there is a bijective function between $$s_1 \cup s_2 \longrightarrow \mathbb{N}$$

kinggeorge · Answer

Since $S_1, S_2$ are countably infinite, there exists a bijection to the natural numbers with each of them. Thus, we can impose some sort of order on both sets such that there is a first element, second element, third element, and so on. Then, since they are disjoint, $S_1 \cup S_2$ contains no elements that are in both $S_1$ and $S_2$. 

Thus, we can impose a similar order on $S_1 \cup S_2$. In particular, let the first element be the first element of $S_1$, the second element be the first element in $S_2$, the third be the second element of $S_1$, and so on. In general, the $n$-th element of $S_1 \cup S_2$ will be the ${n+1} \over 2$-th element of $S_1$ if $n$ is odd, and the $n \over 2$-th element of $S_2$ if $n$ is even.

kinggeorge · Answer

You can prove your second question by induction (it should be obvious, but I'll give a hint if you don't know), and the conclusions are still valid if the sets are not disjoint or infinite, but it's slightly harder to show.

anonymous · Answer

thanks a tonne