 So welcome back to another screencast about sets and set operations and this screencast we're going to look at sets that are infinite and try to work with our four basic set operations that we learned in the last video. So many times when we're using infinite sets we want to really make use of both kinds of notation for sets, both the set builder notation and our roster notation. Set builder notation if you remember is where we write a set in terms of the property that all of its elements satisfy, so it tends to be very compact and economical for writing, but it often hides what's actually in it in which case we want to go to roster notation. It's very important to be able to switch back and forth between those two notations because they reveal different pieces of information about our set. So in what, all the examples that follow, let's let the universal set be the set of all natural numbers. Those are the set of all positive integers for us. And A is the set of all natural numbers that are congruent to zero mod four. B is the set of all natural numbers that are zero mod two. And C is the set of all natural numbers that are zero mod three. And I'm not going to be referring back to those definitions all the time and what's coming up, so you might want to write those down and just keep the definitions handy. So let's talk about intersections and unions, first of all. And let's play with what we have. Let's try to compute the set A union C. Now this would be the set of all natural numbers that belong to either A or C. Remember union means or, it's a disjunction. And so let's write down what that means. That would be the set of all natural numbers, everything in the universal set. Such that N does belong to A, so I would need N congruent to zero mod four. Or, the very important word or there, N belongs to C, N is congruent to zero mod three. So I'm looking for positive integers that are either congruent to zero mod four or congruent to zero mod three. And remember just to help you think about this, we're going to write this in roster notation. Now to help us think about this, to say that an integer is zero mod four means that four divides that integer, the integer is divisible by four. And likewise, this here means that three divides N. So we're looking for integers that do either one or the other. So three would be the smallest element of that set would be three. Remember zero does not belong to the natural numbers for us. So the smallest element I would have here is three. That belongs to the set C here. So the next smallest element would be four that belongs to the set A. The next one up, five would not be in this because this is neither five is neither zero mod four nor zero mod three. So the next item in the set here would be six. Seven would not be in that set because it's also neither zero mod four nor zero mod eight. Eight would go in there because that's congruent to zero mod four. And we could keep writing this, we'd have nine, 12 would be next, and 15, 16, and so on. So the roster notation here is helpful to see what's actually in this set. But if you just looked at the roster notation, you might not get a real clear picture about what exactly all those numbers have in common, that's what the set builder notation is for. Now let's flip this upside down and talk about the intersection of A and C. So this would be the set of all natural numbers that are in both A and C at the same time, just like the intersection of two roads. It's kind of on both roads at the same time. That would be the set of all values of n that are zero mod four. Instead of or we have and this time n is congruent to zero mod three. So that would be numbers, natural numbers that are both divisible by four and divisible by three at the same time. That's going to amount to having the integers that are divisible by 12. This time 12, 24, 36, 48, and so forth. Again, zero is not included because zero is not a natural number. So that's intersections and unions of infinite sets. And let's give a little concept check here to see how what we're doing on this. So the number 10 is an element of which one of these sets. And just by way of reminder, the set B here, which we didn't see in the example, is the set of all integers that are congruent to zero mod two. So with that, pause the video and select all answers that apply. There could be more than one in this case. And in fact, there are more than one. Here is one correct answer and here is the other correct answer. So A union B and B union C. And to understand why that is, let's just pick apart the number 10. Now, we know the number 10 is not congruent to zero mod four. It's actually congruent to two mod four. It's also not congruent to zero mod three. It's congruent to one mod three. But it is pretty easy to see congruent to zero mod two. Any even number will be congruent to zero mod two. Now the fact that 10 is congruent to zero mod two means that 10 belongs to the set B, which is the set of all integers, natural numbers, sorry, that are zero mod two. And so since it belongs to B, it belongs to A union B. Because A union B is the set of all elements that are in either A or B, or possibly both. The fact that 10 doesn't belong to A doesn't really matter. It does belong to B and that's all that counts here. So it belongs to the union because it belongs to one of the sets in the union. And likewise, the number 10 also belongs to B union C again because it belongs to B. Now let's talk about differences and set differences. And speaking of that set B, let's look at the difference B minus A, B minus A. That would mean I am taking all the elements in B and removing anything that belongs to A. So let's set that up. That would be the set of all natural numbers such that N belongs to B and that would mean zero mod two. And I want to remove anything that belongs to A. So I need the final remnants here to belong to be zero mod two and not zero mod three, mod three there. Okay, so the set of all natural numbers that are zero mod two, so they belong to B, but not zero mod three. So that guarantees they don't belong to A. So let's think about what that is. Now B is the set of all integer zero mod two, all natural numbers zero mod two. So that's all even integers that are bigger than zero. And we would just need to subtract out any of the ones that belong to A. I just realized I made a mistake here. Let me go back and change that. A is a set of things that are zero mod four, not zero mod three. Okay, so we're going to subtract out anything at zero mod four. So I would be left with a two, that would be left over because two is not congruent to zero mod four. Four would be subtracted out, so that's not there. The next integer that's congruent to zero mod two would be six and that remains because it's not congruent to zero mod four. Eight would be removed, ten would stay there, 12 would be thrown out, 14 would be there, 16 would be thrown out, 18, and I think we get the picture at this point. So these are all the integers that are in B so that they're zero mod two but not in A so none of them are zero mod four. Let's flip that around and we saw in one of the concept checks from the last video that reversing the order of the difference here matters here and it really matters in this case. Let's look at A minus B and what would that be? That's the set of all natural numbers, such that, let me get my sets right this time, N belongs to A and so N would need to be congruent to zero mod four and not belong to B. So N would not be zero mod two, okay? Now I want you to maybe pause the video here and think about what the set is going to consist of. So click the pause button, write down what you think is in the set and roster notation or in some other notation and come back when you're ready. So actually A minus B here is the empty set. There's nothing in this set. And to see this, just kind of think about what A and B mean. If something belongs to A then N is congruent to zero mod four. Which would mean that I could write N equal to four times Q for some integer Q, but that would make N even because four Q is two times two Q. So anything that's zero mod four must also automatically be zero mod two. And so once you subtract out all the zero mod two stuff from the zero mod four integers, you're left with nothing. Nothing is left over and so A minus B here is actually the empty set. Now one last concept check here to give the notion of compliments, which we have not talked about here. Remember C compliment, let's look at the big letter C here is the set. The little guy C here means compliment. Remember that's the set C compliment is the set of all natural numbers N. Such that N does not belong to C. It's just throw everything out that's in C and keep the rest of the universal set. So if you knew that a number M belonged to C compliment, what exactly would that mean about M? Here are your choices here and come back after you pause the video and select. So the answer here is going to be E. And what does that mean? Well, if M belongs to C compliment, that means that M does not belong to C. That's something outside of C. C is the set of all integers that are zero mod three. And so if M does not belong to that, that means that M is not congruent to zero mod three. And that only leaves two possible choices. It's either one mod three or two mod three. And so that's the notion of the compliment using infinite sets. So that shows how you can use these different set operations in different contexts using infinite sets. Remember to switch back and forth between roster and set builder notation for a full understanding. Thanks for watching.