00:01
Okay, so here we have a, disjoint union, b, intersection c.
00:09
We are going to prove that this guy is a, disjoint union, b, intersection a, disjoint union c.
00:21
Okay, well, how can we do this? okay, let's consider an element x belonging to a, disjoint, joint union, b, intersection c.
00:36
Like this, then this being implies that either, so either, okay, okay, i was saying either x belongs to a or x belongs to b, intersection c.
01:03
And well, this being here, x belonging to a to b intersection c is the same thing as x belonging to b and x belonging to c.
01:21
So in particular, we can see that x must belong to a intersection to a disjoint union b and to a disjoint union c.
01:36
Okay, so let me write like this.
01:40
So x belongs to a, to a, disjoint union b, and x belongs to a, disjoint union c.
01:53
So, therefore, x belongs to this guy.
02:09
Okay, perfect.
02:10
So we showed that this guy is a subset of this guy.
02:14
Now let's show the opposite direction.
02:17
So here we have x belonging to a, disjoint union b, intersection, intersection.
02:24
Section a disjoint union c.
02:31
So this thing implies that x belongs to a disjoint union b and x belongs to a disjoint union c.
02:46
Okay, now let me rewrite these two claims here.
02:52
The first one is equivalent to x belonging to actually we need to write it like this this time either x belongs to a or x belongs to b.
03:20
Okay, perfect.
03:22
And well, similarly, here we have that this statement is equivalent to either x belongs to a or x belongs to c.
03:43
Okay, perfect...