Basic set operations
Bringing the set operations together
Let's now use our understanding of some of the operations on sets to get some blood flowing to our brains. So I've defined some sets here. And just to make things interesting, I haven't only put numbers in these sets. I've even put some colors and some little yellow stars here. And what I want you to figure out is what would this set be, this crazy thing that involves relative complements, intersections, unions, absolute complements. So I encourage you to pause it and try to figure out what this set would be. Well, let's give it a shot. And the key here is to really break it down, work on the stuff in the parentheses first, just as you would do if you were trying to parse a traditional mathematical statement. And then it should hopefully make a little bit of sense. So a good place to start might be to try to figure out what is the relative complement of C in B. Or another way of thinking about it is what is B minus C? What is B if you take out all the stuff with C in it? So let me write this down. The relative complement of C and B or you could call this B minus C. This is all the stuff in B with all the stuff in C taken out of it. So let's think about what this would be. B has a 0. Does C have a 0? No, so we don't have to take out the 0. B has a 17. Does C have a 17? Yes, it does. So we take out the 17. B has a 3, but c has a 3. So we take that out. B has a Blue. C does not have a blue. So we leave the Blue in. So let me write down-- we leave the blue in. And then B has a gold star. C also has a gold star. So we take the gold star out. So the relative complement of C in B is just the set of 0 and this Blue written in blue. So let me write this down. Let me write that down. Now, it gets interesting. We're going to take the absolute complement, the absolute complement of that. So let me write this down. So B-- the absolute complement of this business is going to be-- let me write it this-- the set of all things in our universe that are neither a 0 or a-- and I'll write it in blue-- or a Blue. That's the only way I could describe it right now. I haven't really defined the universe well. We already see that our universe definitely contains some integers, it contains colors, it contains some stars. So this is all I can really say. This is the set of all things in the universe that are neither a 0 or a Blue. So fair enough. So we so far we figured out all of this stuff. Let me box this off. So that is that right over there. And now we want to find the intersection. We need to find the intersection of A and this business. Let me write that down. So it's going to be A intersected with the relative complement of C and B and the absolute complement of that. So this is going to be the intersection of the set A and the set of all things in the universe that are neither a 0 or a Blue. So it's essentially the things that satisfy both of these that has to be in set A and it has to be in the set of all things in the universe that are neither a 0 or a blue. So let's think about what this is. So the number 3 is in set A and it's in the set of all things in the universe that are neither a 0 or Blue. So let's throw a 3 in there. The number 7, it's in A and it's in the set of all things in the universe that are neither a 0 or a Blue. So let's put a 7 there. Negative 5 also meets that constraint. A 0 does not meet that constraint. A 0 is in A but it's not in the set of all things in the universe that are neither a 0 or blue because it is a 0. So we're not going to throw 0 in there. And then a 13 is in A and it's in the set of all things in the universe that are neither a 0 or a Blue. So we could throw a 13 in there. So we've simplified things a good bit. This whole crazy business, all of this crazy business, has simplified to this set right over here. Now we want to find the relative complement of this business in A. So let me pick another color here. So we want to find the relative complement of this business in A. And I'll just write out the set-- 3, 7, negative 5, 13. Actually, let me write out both of them just so that we can really visualize them both right over here. So A is this. It is 3, 7, negative 5, 0, and 13. And I could write the relative complement sign. Or actually, let me just write m-- well let me write relative complement. I was going to write minus. And so in all of this business, we already figured out, is a 3, a 7, a negative 5, and a 13. So it's essentially, start with this set and take out all the stuff that are in this set. So this is going to be equal to-- so you see we're going to have to take out a 3 out of this set. We're going to take out a 7. We're going to get a negative 5. And we're going to take out a 13. So we're just left with the set that contains a 0. So all of this business right over here has simplified to a set that only contains a 0. Now let's think about what B intersect C is. These are all the things that are in both B and C. So this is going to be B intersect C. Let's see, 0 is not in both of them. 17 is in both of them. So we'll throw 17 in there. The number 3 is in both of them, the number 3 is in both of them. Blue is not in both of them. The star is in both of them. So I'll put the little gold star right over there. And so that's B intersect C. And so we're essentially going to take the union of this crazy thing-- which ended up just being a set with a 0 in it-- we're taking the union of that and B intersect C. We deserve a drum roll now. This is all going to be equal to-- we're just going to combine these two sets. It's going to be the set with a 0, a 17, a three, and our gold star. And we are-- I should make the brackets in a different color-- and we are done.