This video is an introduction to sets and that includes both vocabulary and notation. Before we begin working with sets, it's important to understand what a set is. So a set is simply an unordered collection of objects.
So for instance, if I wanted to tell you that I have three brothers, I could say The set of people who are my brothers include Eric, Adam, Kevin, in no particular order. So I didn't write them youngest to oldest, oldest to youngest, favorite to least favorite, etc. Just unordered.
An element is simply an object in the set. So the elements of set B are Eric, Adam, Kevin. And notice they're just separated with a comma. And then, of course, we have the notation here.
So this means is an element in the set or is contained within the set. So I could say Adam belongs to set B. Larry.
does not belong to set B because I don't have a brother named Larry. So it means is contained in or is an element in and of course not contained or not an element in. There are several sets that you should know and it's very possible that you already know all of these but I'm going to go through them anyway just to be safe.
So the first set you should know is the set of natural numbers. The set of natural numbers just the counting numbers. So if I wanted to start listing them it would be 1, 2, 3, on to infinity. W represents whole numbers and whole numbers are 0 and the natural numbers, which means same set but starting with 0 instead. Then we have z, which represents the integers, and the integers are just the positive and negative whole numbers.
And of course there is no negative zero, but we get the idea that it would be onto negative infinity, negative 2, negative 1, 0, 1, 2, onto positive infinity. Then we have the rational numbers. which you would think would be an r, but obviously we have saved the r for something else.
Q represents rational numbers because a rational number is a quotient of the two terms a and b, where a and b are both integers, b is not zero because of course we cannot divide by zero, and a over b is in lowest terms. Then we have r, which represents our real numbers, and the real numbers is everything above it. So all the natural, all the whole, all the integers, all the rational, those are all real numbers.
So anything that is not imaginary is real. And that brings us to c, which represents our complex numbers, and complex numbers are numbers written in the form a plus bi where a actually is a real number it's a the real component of a complex number but as you can see bi that i would represent the imaginary portion of that number so that's all the sets you should know one more thing i do want to point out sometimes you're going to see something like z with a plus or z with a minus and that would just represent the positive integers or if you're negative it would be the negative integers. Now let's talk about different types of set notation. So there's roster notation that is very straightforward. Roster notation just like the roster of a sports team simply listing the elements in the set.
So you can see set S I have said is 1, 2, 3, 4, and 5, so roster notation would just be listing those. Roster notation needs to be used with discrete sets. And what do I mean by discrete sets? Well, hopefully we know since here we are in discrete math studying discrete objects. Discrete sets include things that are countable.
They are not continuous. So let me give you an example of something that would be not discrete, would be zero is less than or equal to x is less than or equal to one. If I'm looking at all of the values between zero and one, there are quite a few. There are actually infinite number of values between 0 and 1 because I could say 0.1, 0.2, 0.3, 0.4, etc. But then I could say 0.11, 0.21, 0.31.
You get the idea. I can just keep adding another decimal place and that gives me a whole other bunch of numbers that can be included. So this is a continuous set and therefore I cannot use roster notation with. that set.
It has to be a discrete set. Then we have set builder notation. Now set builder notation can be used for any kind of set.
So it could be a continuous set or it could be a discrete set. So let me show you an example. Still using set s, if I wanted to write set s in another way, I would say set s is all of the elements x such that, so this little line here just means such that, and everything that follows the such that line is describing the elements in the set. So in this case, I could say all of the elements in the set such that x is an element of the natural numbers, remember the natural numbers were just the counting numbers, and also that x is less than or equal to 5. Does that describe my set?
Well, absolutely it does. It describes my set, but also I could have said all of the elements x such that x is an integer. Well, now integers include the negative and positive values. So here I would have to say that 1 is less than or equal to x is less than or equal to 5. So that's another way.
to correctly talk about set S in set builder notation. Now let's talk about interval notation. So I'm going to start with set builder notation and we're going to move to interval notation. We have to have a different set now because interval notation is what we're going to do with these continuous sets.
So we have to use that interval notation or you can use set builder, but most often interval is the best. It's the clearest. It's the easiest. So let's take a look.
Let's say set B is equal to all of the X's such that 0 is less than or equal to X is less than or equal to 1. So it's including all of the values between 0 and 1 inclusive. So 0 is included and 1 is included. So if I'm looking at a number line where I have 0 and 1, and this might bring you back to middle school days. If I were to graph on a number line all of the values that are included in set B, 0 would be included, 1 would be included, and all of the values between 0 and 1 are also included. Now, if I want to rewrite this in interval notation, I'm going to pay special attention to the endpoints of my interval and whether or not those endpoints are included.
So if I'm looking at the endpoints, 0 and 1 are the endpoints. So 0 comma 1. And interval notation, again, is only for continuous sets. So I don't have to say that it's continuous. Interval notation implies it's continuous.
Now, 0 is included in this set. So I'm going to use what's called a closed bracket. And 1 is also included in the set. And again, because these both say or equal to.
And so that's how I would write that in interval notation. 0 comma 1 with the brackets on the outside which are closed brackets. Now what if instead I said let's let set m represent all of the x's such that 0 is less than x is less than or equal to 1. So what did I change here? Well I still have 0 and 1 but now notice 0 is not included because it's not zero is less than or equal to x, which means zero is not included. There we would use an open bracket.
Now I've still included the 1, so the 1 would still be closed. Or of course, I could have said that 0 is less than x is less than 1, and then I would have open brackets on each side. Let's take a look at two special sets.
The first is not a term that you're going to hear often, but the concept is important. Term or concept is the universal set and the universal set is essentially the set of all elements Under consideration which essentially just means what am I studying? So let's say for instance, I was going to create a Venn diagram and I'm interested only in the natural numbers so This box would represent the universal set of natural numbers. That's how I would denote that.
Hopefully, you've dealt with the Venn diagram before. Anything that goes in this box is a natural number. Let's say then, for instance, I also had a subset, which we're going to talk about more in just a little bit.
Let's say the subset is all of the even natural numbers. Where would the number 1 go? Number 1 is a natural number.
It's a counting number. So it would go within the universal set of natural numbers, but it wouldn't go in here where I'm only containing the even naturals. But I would put the number two.
Number three would go out here somewhere. Number four would go inside. Number five would be out here somewhere.
Number six would go inside. You get the idea. The box itself is the universal set, and that's the set containing everything under consideration. And then, of course, we'll talk more about subsets in a little bit. Then there's the empty set, and the empty set is simply a set with no elements.
And you'll see that denoted sometimes as a zero with a line through it like this, or you might see the set brackets, but with nothing inside, which of course be the empty set. What I do not want to see you do is give me the set brackets around the empty set because that would imply that this is a set containing one element and that one element is the empty set so don't do this coming up next we're going to explore some set relationships