00:01
In this problem, for parts a and b, we're given the universal set u consisting of the integers from 0 up to 9.
00:09
For part a, we're going to pick two subsets, a and b, of this set and show that the complement of their union is the intersection of their complements, where i'm using the right side up u for union and the upside down u for intersection.
00:22
For part b, we'll do the same thing, but choose different subsets, a and b.
00:27
Part c, we want to show that in general, the complement of the union of two sets is the intersection of their complements.
00:35
So let's start with the given universal set, and let's pick two subsets of that set.
00:41
And let's see what happens when we do the calculations.
00:44
We'll come down here.
00:44
Let's make a little bit more room to work.
00:48
So for part a, let's just let a be the subset consisting of 0, 1, 2, and we'll let b be the subset consisting of 1, 2, 3.
01:06
All right.
01:07
So that means the union of a with b is the set 0, 1, 2, 3.
01:18
So the complement of their union, looking back at the universal set, 0 through 9, is the set of all elements in the universal set that are not in this union.
01:29
So we start with 4 and include all the integers from 4 up to 9.
01:37
So here is the complement of the union of those two sets.
01:46
Now, let's take the individual complements of the sets.
01:50
A complement consists of everything in u that is not in a, and b complement consists of everything that is in u but not in b.
02:10
And we want to take the intersection of those two sets.
02:16
So the intersection of the complements will be the elements 4, 5, and so on up to 9.
02:27
So we see for our first example that we do in fact have that the complement of the union is the same as the intersection of the complements.
02:41
All right, so far so good.
02:42
Now let's do it again.
02:44
We picked fairly small subsets a and b for part a, so let's pick bigger subsets this time.
02:52
For part b, we'll let a equal almost all of the universal set.
02:58
We'll let a be the elements going all the way up to 8.
03:01
We will leave out 9.
03:07
And then for b, we will leave out 0 and take all the other elements 1 through 9.
03:18
So here are our two sets this time.
03:23
So the union of these two sets is everything in the universal set because we get 0 through 9 in the union.
03:42
So what does that leave for the complement? the complement of the union, therefore, is the empty set because the union contains everything in the universal set.
03:57
So there's our first calculation for part b.
04:00
Let's see if we also get the empty set when we take the intersection of the complements.
04:07
The complement of a is simply the one element that we left out, namely 9.
04:13
And the complement of b is the one element that we left out of b, which is 0.
04:20
And we see that the intersection of the complements is also the empty set.
04:30
So in part b, once again, we have shown for these two subsets of u that the complement of their union is the intersection of their complements.
04:40
All right, so we're feeling fairly confident that maybe this property holds in general...