Introduction to Set Theory 127 - 1.15

This is going to be the first of a series of videos about set theory, and the main thing I want to do in this video is just talk about the definition of a set and what a set is. So sets are usually defined as collections of the objects. Those objects could be letters, numbers, colors, uh... baseball teams,... more or less anything you want... students in a class... And sets are usually designated with capital letters. So I could write a sentence: "E is this set of all even natural numbers." It would be the collection of all those even natural numbers. Now there are different ways to describe this in math notation, and one of the ways is what's called the roster method, where you just list the elements that are in the set. So I could say that E equals... and then, inside curly braces, 2, 4, 6, 8, and then an ellipsis, the three dots, which means that the sequence is going to keep going. Now, the order of elements in the set is not important. So if I have, let's say, set L, and set L contains the elements a, b, c, and d, that would be exactly the same as if I said that set L contains the elements d, c, b, and a. So the order of the elements in the set really doesn't matter. I'm never going to repeat the same element twice. I'm not going to write something like L equals a set with the elements a, b, c, d a set with the elements a, b, c, d and a again. So elements will not be repeat twice, but the order doesn't matter. Now there are what is called well-defined sets. So I can say something like C is the set of all state capitals. okay well Well, we know there are fifty states and we can make a list of all fifty state capitals. And everybody would have to agree on the same list. But if I said that W is the set of all great writers, well, there's probably no way I can make a list of all great writers and have everybody agree with me. So the sentence "W is the set of all great writers" does not give me a well-defined set. So we're not going to use sentences like that when we're dealing with set theory. A few more things... Ii said earlier that sets are usually designated with capital letters. There are some which are used frequently, and the letters that we use for them are more or less standard. So if you want to talk about all natural numbers, usually the set of all natural numbers is going to be shown by a capital N. If you want to show all real numbers, you use a capital R. There's something called an empty set. If you have a set with no objects it, it's an empty set, and we can show that empty set by writing a big circle with a slash through it. Another way to show the empty set would be to just put those curly braces that hold the elements, the members of the set, and don't put anything inside the curly braces. And then there's something called universal set. The universal set is a set of all the things in general that we're talking about. So if we wanted to talk about the students in school, that would be our universal set. and then we would have subsets of that. We might have all the male students or all the seniors or all the students who were taking biology. But if we're talking in general about that whole set of students, the collection that we're talking about in general could be designated as the universal set, and we use a U for that. So, that's about it for this video. I'm gonna follow it up, and I want to give you a list of the topics in future videos, in case you want to look for them. After this, I'm going to go on to something called set-builder notation, which is an alternate way of describing sets. Then I'll go on to something called equal and equivalent sets, ways to compare two sets. I'm gonna talk about subsets and proper subsets. So this would be like if the universe that you were talking about was a school, then all students in biology might be a subset of that universe. I'll talk specifically about the universal set and something called set complements, and then we're gonna get into the actual things we do with sets. So there are set operations, and set operations are very often described or visualized using Venn diagrams. And, finally, there'll be some videos about solving word problems with Venn diagrams. So stick around. I think you'll find plenty to watch. Take care, I'll see you next time.