00:01
So we're given countably many sets an, which are countable, and we want to show that their union is still countable.
00:12
So the countable union of countable sets is still countable.
00:16
So to do that, let's write a diagram where we have in the first column the elements of the set a1.
00:24
So let's call them a1, a2, a3, and so forth.
00:28
Then in another column, we write the elements of the set a2.
00:31
Let's call them b1, b2, b3 and so forth.
00:34
Then another column, the sets of a3, c1, c2, c3.
00:37
And we completed the diagram in this way.
00:41
Now, here, let's assume that all these sets a, they're all infinitely countable, and all their elements are different.
00:52
Because that's the hardest case.
00:55
Because if some of these sets a were finite, then that's not a big deal.
01:01
The union of finite sets is still finite, so that doesn't change.
01:04
Anything and if there are some overlapping between the sets so if an element is both in some a1 and a 2 well when when you're in them the union is smaller than it would be if all the sets were different so the set where all the set the case where all the sets are infinitely countable and all this joint is the hardest possible case so let's assume that so all these a 1 a 2 b3 and all these numbers are all different...