Problem two: Prove that for sets A, B, C in some universe we...

Problem two: Prove that for sets A, B, C in some universe we have $(A \times B) - (A \times C) \subseteq A \times (B - C)$ Hint: You know that to get a subset proof going you assume you have an element on one side and show that it must be on the other. Here, since we are dealing with the cross product, each element is an ordered pair. Start one by assuming $(x, y) \in (A \times B) - (A \times C)$, apply the definition of set difference to pick that apart, and apply the definition of cross product.

Transcript

00:01 So in this question we are asked to prove something basically.

00:05 So we want to assume that all sets are subsets of some universal set u.

00:11 And the statement is for all sets a and b, again, these are subsets of some universal set u.

00:23 If a complement is a subset of b, then a union b is equal to this universal set u.

00:36 And they give us a hint says that once we've assumed we have a and b are any sets with this property.

00:45 So what are we supposed to show in order to actually show this equality of sets? so i'm going to write this in red here.

00:55 Basically what it means to have an equality of sets is that the set on the left of the equation, a union b, needs to be a subset of the set on the right.

01:09 Side of the equation.

01:10 So first of all, a union b must be a subset of u and at the same time, we need u to be a subset of a union b.

01:24 So it's just that you know you have two sets, right, one on either side of the equal sign and we need to share set containment sort of both directions.

01:34 So the left thing should be a subset of the right thing and the right thing should be a subset of the left thing.

01:41 So those are the two things that we should show.

01:44 And to actually do the proof then, okay, so let a and b be any subsets of this universal set u, such that, right, they have this special property that a complement is a subset of b.

02:14 So first of all, since a is a subset of u, right, by hypothesis, and b is also a subset of u, of course, a union b must be a subset of you.

02:32 Right? a union b consists of elements that are from a or b or both, but certainly any element that's in a or b is also in you, right? because each of those individual sets is in you.

02:46 So that's sort of the easy direction.

02:48 It's obvious.

02:50 You almost don't need to write down, but we should just to be, you know, careful.

02:56 So we have what i just want to line blue, we have the sort of forward direction.

03:01 We've got the a union b is a subset of you.

03:03 We've shown that part.

03:04 Now i'm going to show the other direction that u is a subset of a union b.

03:09 So i'm going to just write in an extra sentence.

03:13 This is just for the sake of explanation.

03:15 You don't have to include this in the proof, but we want to show that u is a subset of a, union, b...

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