00:01
We want to prove that if we have three sets a, b and c, then the union of a and b minus the set c is equal to the union of the sets a minus c and b minus c.
00:20
So the proof the equality of two sets.
00:23
We prove the two contentions.
00:29
That is first, we prove that a union b, minus c is a subset of a minus c union b minus c let's say in order not to be with letter c let's use a red subset symbol so what we want to prove in the first part is that this set is a subset of this one and for that we take any element in the set a union b minus c and proves that it is also an element of the set on the right.
01:32
So let x be an element of a union b minus c.
01:40
What does it mean that x belongs to this set? it means that it belongs to this first set, this one here between parentheses, but it's not in the second.
01:53
So x belongs to the union of a set.
01:56
And b, but is not in c.
02:03
That's the definition of the difference of two set.
02:09
Okay, but, but now, x belongs to a union b means that x belongs to a or belongs to b.
02:36
That's the definition of the union of the set.
02:39
An element is the union of the sets.
02:42
If that element is in one or both of the sets.
02:50
So it can be here or here.
02:55
If the case is that x belongs to a, then since it is true that x does not belong to c here, we have that x is in a but not in c.
03:19
That is a, so that is x belongs to a minus c by definition.
03:25
Because x is in the first set but many second.
03:31
And then x belongs to the union of a minus c and b minus c because belongs at least to this set.
03:43
So we started with an element in the set on the left here and prove that is in the set on the right here.
04:03
But we have another possibility.
04:06
On the other hand, if x belongs to b, because remember, at this step here, we have two options.
04:23
X belong to a or belong to b.
04:25
So we started the case when x belong to a.
04:29
And we proved that in that case, x belongs to the set on the right of the contention symbol here.
04:39
Okay...