 So welcome back to another screencast and let's do another example where we're proving something about sets using this notion of the quote unquote algebra of sets where we're using set identities to chain together equal sets rather than the choose an element method. Both methods are okay. Some work better in some situations than others. So we just want to have two different methods in our tool bag here. So we're going to prove this proposition here that says for any sets a, b, and c that are subsets of some universal two, we have the set a minus c and the set c minus b are disjoint. Remember we defined disjoint sets to be sets whose intersection is empty. So we're going to work with that. And in an earlier video what we said was that oftentimes if we're proving two sets are disjoint we do this by contradiction. We suppose that they are not disjoint and so there's something in their intersection and then derive an contradiction from that. But we're going to do this differently now using the algebra of sets. So let's go set it up. So we're going to show that a minus c and c minus b are disjoint and that is to say that the intersection of a minus c and c minus b is the empty set. That's our eventual goal. So instead of doing the choose an element method or contradiction we're going to use algebra of sets here which means we're going to start with the left hand side. Just treat it like any old equation. Let's start with the left hand side and do some valid mathematical steps which include our set identities as seen in those two theorems we saw. And eventually you have a chain of equal signs that would start here and go down and eventually equal the empty set. That would be great. So let's see if we can make it happen. Well one thing we can definitely do here when I'm looking at the left hand side I'm looking mainly almost like algebra again looking inside the parentheses and the order of operations. So what could I do inside the parentheses? Well I have this one fact identity about sets that we actually proved in one of the earlier videos that the difference between two sets is the same thing as saying the intersection with a complement. So a minus c is the same thing as a intersect c complement and I'm intersecting that with c minus b which I can write as c intersect b complement. And the justification for that if you go back to the actual theorem statements this is called the quote unquote basic property one of the quote unquote basic properties. We proved this earlier but if you're looking at the theorems from previous slides it's called basic property number two. If you want to pen a name on it. Now what can I do? Well if you notice what I have here is a bunch of intersections. I have one intersection here, another here, another here. Why don't I regroup that? I can use the associative property here. I'm in quite a bit of room here and I'm just going to regroup those middle two, especially get those middle two intersections together. So I'm going to rewrite this as a intersect parenthesis c complement intersect c parenthesis parenthesis intersect b complement. And again this is by the associative law, associative law that allows me to regroup intersections however I want. Well now what can I do? The reason I was doing this is because I saw c complement and c kind of together. Now that I really have them together just think about what is c intersected with its complement. Well that ought to be the empty set actually and it is. So this is, I can replace the innermost parenthesis here with the empty set. And if you want to rule for that, that's actually one of the things we proved. This is a proposition that we've proved back in screencast 531. Which is why we brought it up because I knew we were going to be needing it a little further down the road. That's actually not in your book but we proved it so it's now an identity that we can use forever. So that's why that's true. We proved that result earlier using the choose an element method. So like I say again you do need to keep both methods in mind because sometimes one method will work better than the other one. It just depends. Now let's see what we got. Well here this seems suspicious. I'm intersecting a with the empty set. And I can say definitely that that equals the empty set itself. And the reason I can say that is because that's a proven set identity. That is one of the properties of the empty set and the universal set that was in the theorem 5.18. And then finally I can use that property one more time to say this intersection is the empty set. Again it's the same justification. And so if you look and just follow the chain of equal signs I got what I wanted. I started with a minus c intersect c minus b and that's equal to this by one of the previous properties we saw. That's equal to this by the associative law. That's equal to this because of our proposition we proved earlier. That's equal to this because one of the properties we saw earlier. And that's equal to finally the empty set again because of the properties here. So I'm just using things that I know already about sets to just chain equal sets together. And ultimately I have this equation up here which is what I wanted. So algebra sets pretty powerful. It does require you to have a pretty good memory. And sometimes you have to go invent a property and go prove it on your own. But if you can make that happen then these proofs work out very nicely. Thanks for watching.