00:01
In this problem, our job is to determine in each case whether the given set of subsets is a partition of the set a.
00:07
So we recall that the set a in this problem consists of the integers 1 through 6.
00:12
We also recall the definition of a partition.
00:16
A partition of a set like a is a set of subsets of a that have to have the following two properties.
00:37
So we will call those subsets a sub i's.
00:41
And what two properties must they satisfy? first of all, the union of all those subsets has to equal a.
01:01
So you can't leave any elements out.
01:07
And if you take any two subsets in the partition, their intersection is empty.
01:17
So as long as you take two different subsets and compare them, they will be disjoint.
01:23
So any partition is a set of subsets that must satisfy both these properties.
01:27
No element gets skipped.
01:29
That's the first property.
01:30
And no element gets listed twice.
01:32
That's the second property.
01:35
All right.
01:36
So we were given three possibilities here, three sets of subsets.
01:41
We'll check those properties in each case.
01:44
In the first case, the first potential partition is the set of subsets of a consisting of three subsets.
01:51
The subset 1, 2, the subset 3, 4, and the subset 5, 6.
02:03
And so what we note is that if we refer to these as a1, a2, and a3, then we can see, first of all, that if we take the union of those three subsets, we get everything in a.
02:19
We get 1 through 6...