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Introduction to Sets in Discrete Math

A free video tutorial from Grant Hall
Credentialed Actuary and Dedicated Math Tutor
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Introduction to Sets

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Master Discrete Mathematics: Sets, Math Logic, and More

Master Discrete Mathematics: Learn and master all of Discrete Math - Logic, Set Theory, Combinatorics, Graph Theory, Etc

06:49:41 of on-demand video • Updated October 2020

Most Complete Course on Discrete Math offered on Udemy
Fully Understand Mathematical Logic
Grasp the Complex Topic of Counting for Advanced Mathematics
Know When to Use Combinations or Permutations
Solve Real World Problems Using Combinations and Permutations
Discover How to Solve Mathematical Proofs Using a Variety of Methods
Learn Mathematical Notation to be able to Understand Complex Proofs
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Hello and welcome to today's video, which is one of the most important videos in the course, and that's the introduction to sets a set is a collection of unordered, distinct objects that we call elements. And there's really three key things here. The first is that they're unordered. We'll talk about what that really means in a second. They're distinct, so they're unique. So each number or each word in the set is not like the others. And they also can be numbers or they can be something else that's important as well. They can be numbers. They can be words. They can actually be sets inside of the set. There's all kinds of different things that we can have inside of sets, and there's a lot of different ways that we can represent sets in mathematics. There's a one way that's good for visual learners will go through that. Right now, that's where you kind of consider is a box. So you have this set, we'll call it set a and it could be a box with how about the number five, the number two and the word car. So this is just to show you that there can be words and numbers inside of a set, and that's OK. And we label this set a so that's another way we could write it. We could say this is set a we'll talk about that in a second. And then there's kind of the most mathematical way. This is the way you're going to see in this course most often. So you put a little curly brace, you write all the elements out, so five to in car and then you put it in another curly brace there and you just separate them by commas. So that's another way to write and that's fine, too. And that's how we normally do it. So Curly braces each element separated by commas. And then also we could just say set a.. So if we define set a to be the set that contains five two in car, I can just write in a problem the set A minus the set B or set A. What's the power set of that? Could be a question and you'd have to know. OK, well A is the set five two in car and you can visualize it either way. So that's just the introduction here and that those are some of the ways that we can represent the set. Now let's get into these these important points. So remember that I highlighted the three words. I said unordered, distinct, and anything can be inside of them. Well, now let's talk about that more. So first, there's no order in a set there. Unordered, which means the set zero one five is the exact same thing as the set one five zero. And the best way to think about that is to think about it using the visual representation where we have the box. Right. We have the box and we'll say we had one five zero. But now if I have one or zero one five, notice that these are the exact same thing. It doesn't matter where it is inside of the box, it just matters that in my box I have a zero, I have a one and I have a five. That's all the set is saying. So there's no order in a set. We got that the second the repeated elements are only listed once. That was when we highlighted the distinct words. So you're never going to see in a set two of the same element. You're never going to see zero zero or one one, so on and so forth. You're only going to have it listed one time. So the set containing zero zero zero one five five, we would just write a zero one five. We don't say how many of there how many objects there are. That's really important and something you got to know. And also that's can be finite or infinite. Remember, finite just means it's countable. So that's something like zero one five. It's obviously finite. There's only three numbers, but they can also be infinite. So look at this set Z. This is infinite because we have those dotted lines. That implies a repeated pattern. And what is this pattern? Well, we're going from negative to negative one zero one two. It's going to go all the way up to like three, four or five, six and all the negative side is going to do the same thing. Negative three, negative four, negative five, so on and so forth. Well, these are actually the integers, aren't they? All the integers, negatives and the positives, and they include zero. And that's an important set that we're going to talk about later. And we usually represent in mathematics by Z, so that's an important one we'll get into. OK, so here are the important sets. First, the natural numbers. This is the set of positive integers. Some teachers like to say the natural numbers include zero. They'd like to say the set of natural numbers is zero. One, two, three. But the vast majority are doing what we do here, which is the one, two, three, four. And I like to think of natural numbers as counting numbers. That's what I like to say to myself because it just makes a little easier. So counting numbers is the same thing as natural numbers noted by the set in and counting numbers makes sense that it's one, two, three, four or five, because when you learn to count to ten, you don't usually say zero. You started one. You say one, two, three, four, five, six, so on and so forth. So I like to say to him is counting numbers and that's represented by the letter N, so that's important because in problems, especially in discrete math, they're going to say set in and they might say, what's the carnality of set in or something like that or is set in a subset of this set. So you need to know what it means. And that is the set one, two, three, four and all the way to infinity and also set of integers we looked at that one includes all positive and negative integers. I didn't know another word to do that. It's really just all positive and negative like numbers, whole numbers. So all the way from negative infinity. Counting by one all the way to zero and then all the way to infinity, and then we have the rational numbers. So I'm sure you guys know this already there. Those are any number that you can write is a fraction. So rational numbers include things like five halves. How about four thirds? Anything like this, three seconds. And they also include the whole numbers. Like two, like one, like zero. So it's a lot more extensive than the integers. But the problem that we have right now is that we don't have a good way to write this as a set. So let's do our best. And then in the later video, we're going to have set notation that we're that will really help us write this out. And usually the rationals are noted by a AQ, so we'll say Q equals. And now we want to somehow figure out how to write the rationals. And I know that I'm going to do it's going to repeat all the way from negative infinity to infinity. Well, how about we start with just saying, let's see, the thing is, we really don't have a good way, so we're just going to kind of have to guess how about negative three over two? Then we could have negative two over one, negative one over two and so on and so forth. We just have to show some pattern, just showing that it has all fractions. There's not a good way to do it. And that's the problem we have right now. This is wrong. There's no way to show it would just doing a repeating pattern. So what we need is we need some variables and we'll get into how to write this that later on. But just know the rational numbers notated by cue includes any number you can write a fraction. And we'll learn the set notation in a second in a few videos. All right. So now elements in carnality, this is kind of the last thing I want to touch on in this video. It is very important. So hopefully you're still still awake. And the symbol for inclusion is Epsilon. So when I want to say, let me show you here, so let's say I have a equals, I think we've been doing 015. So you'll often see this in discrete mathematics. You'll say zero is an element of a and that's that epsilon right here. That's the inclusion. So I said zero is an element of. Well what about what if I want to show exclusion. Well how about seven. That's not not not an element of AA. So I just put the epsilon and I do dash through it. Seven is not an element of a so it's that simple. And the way we do it is we just say, well, zero is obviously inside the set and seven is nowhere to be found. So that's how we know that it's an element. And we'll get into some more specifics of how to really tell if it's an element, because this will get more complicated later on and then cardinality the size of the set. It's really the number of elements that you have and it's notated by straight lines on both sides. And you might be saying, well, that's how I notate absolute value. And you're correct, it looks the exact same. So it's absolute value of B Well, you wouldn't want to say absolute value. You'd want to say cardinality of B is going to be the number of elements and B equals number of elements in B and looking at said B that we have here that B equals the square rectangle Pentagon. Obviously there's three three elements. Right to the cardinality of this set is three. And if I wanted to show inclusion exclusion, I could say square is an element of B and five is not an element of B, so the carnality is the key thing for here. And that's just saying the number of elements. Well, I count three, one, two, three. So that's the cardinality. So this was just a simple introduction to sets. I hope you kind of understand the basics. The keys are these underlined words that sets our collection of unordered, distinct objects that can be numbers or something else. And we'll get into the set notation, which will help us write the rational numbers that we touched on. And we'll also do a lot more practice with cardinality, because that's something that can also be a lot more advanced than what we just touched on today. So thanks for watching and I hope you enjoy the next few videos. Have a good one.