 Hello and welcome to this screencast on sets and set notation. So sets are very important for us. There are many ways to the primary building block of what we do in mathematics. So let's start off with a definition. We're going to keep a pretty loose definition of what a set is, because it turns out to be fairly hard to give a rigorous definition of what a set is in mathematics. So we're just going to say a set is a well-defined collection of objects. Pick on each of those words. Well-defined means there's no ambiguity in the contents of the set. Things are enlisted twice in different forms. Everything's very explicitly laid out. A collection of objects means that we're dealing with a fairly generic structure here. We could have numbers inside a set. We could have functions inside a set. We could have names inside a set. We could have sets inside a set. Just a collection of objects, just more or less the same way when you pack a suitcase to go on a trip, that's a collection of objects. So here's some quick numerical examples here, just mainly to fix down some notation. This set here consists of four objects, namely the numbers 1, 2, 3, and 4. The way that I'm writing this set here, first of all, notice the curly braces that I'm using to denote the beginning and the ending of the set. That's very common standard notation for how we write sets. And what I'm doing here to specify what the objects are, is just to list them, 1, 2, 3, and 4. And this is a notation we're going to refer to as roster notation, just like the roster of a sports team. We would just be listing out who's on the team. And so we can specify a set just by listing in a comma separated list the elements of that set. Notice again that we don't care necessarily what the order is in this. So here is another way to write the same set, 4, 3, 2, 1. Those two sets are the same. Here would also be the same set if I wrote 1, 1, 2, 2, 3, 4. We don't count duplications at all. We factor those duplications out. So there's a set written in roster notation. Here's another set, slightly longer, 1, 2, 3, 4, 5, 6, 7, 9, 10. That has 10 elements in it. And since it's kind of a pain to write a set that's this long, many times we'll establish a pattern in a roster notation like this and then use the ellipses or the dots here to kind of say the pattern continues all the way up to 10. Here's another example, except this time the pattern continues but doesn't stop 1, 2, 3, et cetera. So that would be the same thing as saying the set 1, 2, 3, 4, 5, and so on. And it never stops. So we're going to use the dots to denote that infinite continuation. This, of course, is just the set of natural numbers, the set of all whole numbers that are positive. So that would be one way to write the set of natural numbers. We also had a symbol, kind of a funny-looking end for that set of natural numbers. And here's the last example here. The third example was a set that had infinitely many elements in it, whereas the first two only have finitely many elements. This last one has infinitely many elements as well, except it's infinite on both sides. So we have a pattern that is continuing, and we kind of see what the pattern is by looking at the middle of the roster notation, and then it continues off to the end. So this is basically all whole numbers at all, otherwise known as the set of integers. So that would be a roster notation way of writing the set of integers. Actually, I lied a second ago, and I have one more example to show you here. Here's a set that would be very hard to write in roster notation, because it would be hard to list the elements. The set of all real numbers between one and five. Now, the set of all integers between one and five would be easy. We would just say two, three, four. We'd have to hag about whether we meant to include one and five. But the set of all real numbers, there aren't just three of those. There are infinitely many of them, and we can't really list them. We have things like pi is between one and five, and square root of three is between one and five, and so on and so forth. So we have this set as infinitely many numbers in it, and they're not easy to list. So roster notation does not work so well for that particular set of numbers. So now that we've seen some examples and got our minds around the notation, let's do a concept check. So what is the set of all natural numbers in that satisfy the inequality n squared less than or equal to four? Here are your five options here. Notice that the fourth one is an infinite set. And E, we're saying that, okay, the set of numbers is infinite, but it's like the last example. It's not easy to write in roster form. So take a moment to read over these five options, pause the video, and make your selection, and come back when you're ready. And we're back. And so the answer here, the set of all natural numbers that satisfy the inequality n squared less than or equal to four is this one at C. Did you say B? You gotta remember what a natural number is. A natural number is a positive integer in our usage of the term in this course. It's not universally agreed upon, but in our usage, natural number is a positive integer and zero isn't positive. So one and two, that is the set of numbers that satisfy n squared less than or equal to four. So one squared is one, that's less than or equal to four. And two squared, that's equal to four. And that's also less than or equal to four. B doesn't work because zero is not positive. A doesn't work because it's got not only non positive things but straight up negative things as well and D is the same problem. So that's a quick overview of sets and set notation. Thanks for watching.