 Welcome to the screencast on set builder notation. So let's have a look at a set that's a little different than one you've seen before. We're going to look at the set M of all motion pictures ever made, and focus on this subset of M right here. Here's a subset of the set of all motion pictures ever made, and we're going to list this in roster notation. Star Wars, Blade Runner, Indiana Jones, the fugitive, cowboys and aliens, dot, dot, dot. Now normally we put the dot, dot, dot there to indicate a pattern is taking shape, and we want to continue that pattern. It may not be totally obvious to you what that pattern is in this case. That's my question. So think about this for a second and test your pop culture knowledge, and what is the pattern that's connecting all of these movies together? What are all the elements of this set have in common? And if you have some pop culture knowledge and know a little bit about movies, then the answer is fairly straightforward, and that is Harrison Ford is an actor in all of these. Harrison Ford is an American movie actor and was a star or the star of all of these guys, including the ones we don't see here. Now which of these two sets are equal? They contain exactly the same elements, but it's a little clearer and a little bit more efficiently written in the second form than it is in the first form. Harrison Ford obviously is not in infinitely many movies, and so I don't mean that the set continues forever, but he's been in a lot of movies, and it would be really hard and awkward to list them all, and we're relying on people's pop culture knowledge to fill in the dot dot dot. So it's better to write this set here, not in terms of a roster of its elements, but rather in terms of the property that all those elements have in common, right? So the set of all movies in which Harrison Ford was an actor, that's a much clearer way and shorter way to write this set than roster notation. We're going to shorthand this even further by introducing some notation here. So we're going to write this set, the movies in which Harrison Ford was an actor, kind of in two parts, and label this part one and part two over here. And notice that they're separated by a vertical bar. The first part of this set here, before the vertical bar, I'm going to state the universal set of things that I want to think about. In this case, I'm saying look at the set of movies that belong to M, okay? Not just sci-fi movies for example, although I think that's all the ones that I listed, but all movies period, so that's the universal set. So I'm going to say the set of all things in the universal set, this vertical bar, we're going to read it off as such that. So the set of all M in the set of movies, such that, and in part two, we're going to list the property that all these elements we want in our set have in common. So it's the set of all M in capital M, such that Harrison Ford was an actor in M, the movie in question. This is a way of notating a set called set builder notation, because we are building the set, not by listing the items that are in it, but kind of trying to isolate the one property that they all have in common and building the set based on that property. So set builder notation, we write a set in terms of its universal set, and then such that, and then list the property that all the elements of the set have in common. Let's look at a few more examples here. Mark Hamill was another actor in the movie Star Wars, of course he played Luke Skywalker, and so you might think, okay, what is the set of all M in the set of movies, such that Mark Hamill was an actor in M? Well, let me put the result up here. Let's try to list these out in roster notation. We're going to convert from set builder to roster. So what's the set of all movies where Mark Hamill, movies now, where Mark Hamill was an actor in M? Well, Star Wars was one, the sequel to Star Wars, which is The Empire Strikes Back, Return of the Jedi, and you know what? You kind of run out of, it's a little hard to list these out once you get past those three. I happen to have IMDB up here and Mark Hamill's entry, and you know what? If you start scrolling down here, you don't see a lot of movies that he's in. I guess there's one that was called Back to the Sea, one that was called Airborne, a ton of video games and TV shows, something called Minkowl. Anyhow, we can keep listing these out as the point here. I'll put that last one here. Minkowl, I don't know what that is, but anyhow, that would be a sort of a partial roster notation that would indicate this set. But isn't it a lot clearer said this way than this way? If I gave you this set and said, what do these sets have in common? I wouldn't have any idea what they have in common, but this one really isolates what they all have in common, and that's how we build the set. Looking at a couple of more mathematical examples, what is the set of all x in the real numbers, such that x squared minus four is equal to zero? Let's list those in roster notation. So I'm looking for all real numbers x, such that x squared minus four is zero. Well, I can take a one of them, that would be two, and there's another one, namely negative two, and that's it. Okay, so that's easy to list out in roster notation. In this case, the roster notation is shorter, although it may not highlight the property so much that I am thinking of. Third example, what is the set of all integers, x is in the integers now, notice the change in the universal set, such that x squared minus four is less than zero. The universal set makes a big difference here. If I didn't have the integers here, this would be a very large set. But since I only want to think of x's that belong to the integers, there aren't too many of these that happen. Negative one would be one of those, because negative one squared minus four is less than zero. Zero would be another, and one would be a third one, and that's it. So we've converted from this set builder notation to that roster notation. Again, the roster notation for a set focuses on literally what's in the set. Set builder notation focuses on the property that all of the elements have in common. And these are just two different notations, and one notation may be more useful for us than another in a given setting. So it's important to be able to go back and forth between the two notations. There's another way to write a set and set builder notation as well that might make things a little bit easier to think about. So take a look at this double infinite set here. Pattern continues on both sides. And you might stare at this for a little bit and think about what is the pattern here. It's a little hard to isolate. It's fairly easy to see once you look at this for a little bit that all these elements are different by 5. They all have a difference of 5. I'll just go ahead and jump to the answer here to write in a set builder notation. These are all integers, so I might say the set of all in the integers such that the thing that all these elements have in common is that they can be written as 1 plus a multiple of 5. This is 1 plus 0. This is 1 plus 5. This is 1 plus 10. This is 1 plus 15. This is 1 plus negative 5. So all of these elements have this property in common that they are integers such that n is equal to, let's say, 5k plus 1 for some other integer, k. That would be correct. It kind of reminds you of our definition of even numbers. It's kind of closely related. This is a little awkward though here. So another way that we can write this in set builder notation is to put the property first. Instead of specifying the universal set and then the property, we can specify the property and then the universal set. So we might also write this set like so. We can write this as the set of all numbers of the form 5k plus 1 such that k is an integer. And that is the same set as both of these, but it's the shortest and most efficient of all. So we can also write, and you will also see written, the property followed by the universal set that contains this variable here that's in the property. So let's end off with a concept check. Here is a set that's given in roster notation. I'd like to think about what it looks like in set builder notation or could look like in set builder notation. So take a look at these five options and pause the video and select all of the ones that are correct. There could be more than one this time. And there are more than one this time, and that would be c and d. Let's focus in on d first. This set consists of all the powers of two, the positive integer powers of two. This is two to the first, four is two to the second, eight is two to the third, two to the fourth, fifth, sixth, and so on. So that's the set of all things that look like two to the k power where k is a natural number. That rules out e because I don't have any negative powers of two here. I don't see two to the minus one or two to the minus two or any such thing. So the universal set really matters here. And it's not a or b because these aren't just multiples of two, they are powers of two. So there are some things that are in, say, b that are not in the set up here, like the number six would be an element of the set here, the set of all two to the n where n is a natural number, but it is not an element of the set up here. So these two guys are not correct either. So that is a look at set builder notation. Thanks for watching.