 An important concept in mathematics is that of a set. So definitions are the whole of mathematics, all else's commentary. We should try to define a set, but it's so basic a concept that it's not possible. And this leads to another important idea in mathematics. It doesn't matter what something is, what matters is what it does, and how it relates to other things. And so we define a set consist of elements, which are said to belong to the set. And rather than defining what a set or elements are, this definition says that this concept of set is related to this concept of element by the idea of belonging to. And let's introduce some notation. If x is an element of a set A, we write x is an element of A, where this is sort of a stylized E. There are two common ways of describing a set. We can use list notation where we list the elements of a set and enclose the whole in a set of braces. So A might be the set whose elements are 4, 8, 15, 16, 23, 42. Now an important idea here is that list elements should not be repeated. We'll see why that's a requirement later on. The other way we can describe a set is to use what's called set builder notation and describe an inclusion rule that is used to decide whether something is an element of a set. So we might say that B is a set of things where what we're looking at is not number. It's important to understand the difference between the two forms of notation. So for example, let A be this set and let B be this set. And let's list the elements of A and of B. The first thing to recognize here is that A is given in list notation. So its elements are going to be the things separated by commas. And so the elements of A are this block of text, the odd numbers between 0 and 10, and this block of text, except 7. Meanwhile, B is in set builder notation, and so its elements are things that are odd numbers between 0 and 10, except for 7, and those are the numbers 1, 3, 5, and 9. So here's an important concept. The empty set designated this way is the set that contains no elements. In list notation, we have our empty set, and we're going to write down everything that's in the set, and once we have that list of everything in the set, we'll throw that list into a set of braces. Now, given any set, there are some standard questions you should always ask. One important question is, does the set contain any elements? So let's consider, let's list the elements of the empty set, this set, and this set. Definitions are the whole of mathematics. All else is commentary. This is the empty set, which doesn't contain any elements. Now, this set consists of something enclosed by a set of braces. So it's evidently in list notation, and the elements of this set are the things inside the braces. This set contains the element, the empty set. And finally, this is also in list notation, and this set contains the element 0. Once we have sets, we might try to compare them. So given two sets, let a and b be sets, we say and write the following, a is equal to b when a and b have the same elements. We write this, and we read it as a is a subset of b when every element of a is also an element of b. Hey, it's useful to make one for the distinction, a is a proper subset of b if a is a subset of b and a is not actually equal to b. And to emphasize this, we sometimes use this notation or even this notation for a proper subset. One other feature, if a is a proper subset of b, we say that b is a super set of a, or that b contains a, and we write it this way. Just as a note, we don't often use this notation or this terminology. So let's consider a couple of sets and let's find the relationship between the sets a, b, c, and d. So definitions are the whole of mathematics. All else is commentary, so we'll pull in our definitions. And we see that b has elements c, d, and e, and d also has those elements c, d, and e. And since they have the same elements, then they're equal. And one of the things that this means, if b and d are equal, but the only difference is the order in which we list the elements, it follows that the order we list the elements in a set is not important. Now if we look closely, we see that every element of b is an element of a, but a is not equal to b, then b is a proper subset of a. How about c? Well, we might notice, first of all, that c is not equal to a and not equal to b. Not everything in c is in a, so c is not a proper subset of a or of b. And in fact, we might observe that while c has some elements in common with a, it's not equal a subset or a superset. And so you might ask the question, what is it? We'll take a look at that next.