So in this lecture we are going to talk about subsets of real numbers. Just some specific subsets of real numbers. They will be quite useful in illustrating some of the operations on subsets that we talked about last lecture.
For us, it's quite... we will take a very simple approach to thinking about the real numbers and we will be thinking of them just as numbers that have a decimal expansion. The important thing about decimal expansions is that they allow you to compare to real numbers so you can't always do this with a number system so if you had the complex numbers you can't say whether one complex number is bigger than another one.
But with real numbers you can, and the reason you can do that is because they have a decimal expansion. And so to compare whether two positive numbers are the same, one is bigger than another, you look down the decimal expansion and you find the first time that one of them is bigger than another one. And that, apart from some... some very specific cases where you've got.999. That's the way that you can tell whether one is bigger than another.
So we have this idea of less than or equal to, or less than. And that's how we're going to construct some of the subsets of the real numbers. And these are going to be called... intervals, so I'll just go straight in, if a and b are real numbers, then the closed interval, closed interval, written a comma b, this is the set. such that x is a real number and we want a to be less than or equal to x, so x somewhere between a and b, but it can be, we're including the endpoints here, so x is less than a and x is less than or equal to b.
And we sometimes write that. in the following way we just put the two of these inequalities together because it just saves a little bit of writing okay let's use one of these a couple of these closed intervals together with an operation and we can see what we get when we do an intersection so eg if i take 0, 2. That's all the real numbers that are between, and we can draw them on a number line, are between 0 and 2. I can think of those as being that one there. That could be my set A. And we take the numbers. the interval 1 3 and that can be my set B.
There's a 1 in there and I can take my B here. I'll go in the other direction for that. This is B. Okay, and I can already So what about what I draw in here? It's actually a Venn diagram.
I can use a Venn diagram to form some unions and some intersections and see what I get. So take 0, 2, intersect 1, 3 and I'm going to get all the stuff that is in the both of them and I can see that that's going to be the stuff that's between 1 and 2. So that's actually going to be the closed interval 1, 2. If I take the union then I'm going to get all the stuff that's between zero and three. I could take the symmetric difference if I liked and then I would be getting all the stuff that is between zero and one but not including one and similarly between two and three. But before I do that I'm going to describe what open intervals are because that will give us a better way of talking about those. So if a, b are in R Then the open interval and I would write it as I have done in the notes with a Curly we're not curly but a rounded bracket as so But you will find it also with this notation so where you just put the you reverse the square brackets there and I think maybe on the course works it's described like that but I like this notation but it's just exactly the same thing as we had before but we're not going to include the end points so a and b is going to be the stuff in real numbers such that a is less than x but not including the equals, so it has to be strictly bigger than x, and x is strictly less than b.
So that is my open interval, and you can think of it as being a bit like, so if I do 1, 2, it's sort of like where I have everything almost getting up to a 1, but not quite there and everything almost only out of two but not quite there and then everything in the middle okay so um with the uh with the open interval i can now describe what the symmetric difference is here because if i want to be in a or b but not both of them then i want everything going up from 0 to 1 but I can't include 1 because it's in B so everything all the way up to here but not quite here and everything all the way up from 2 to 3 but not quite including 2 so for that I'm going to have if I want 0 2, a symmetric difference of that with 1, 3, then I'm going to get all the stuff from 0 coming up to 1, but not quite 1. Union, all the stuff that goes from 2 up to 3, but not quite including 2. So here I kind of mixed the notation so the sort of open and closed interval a b here is just the x in r such that a is less than x and x is less than or equal to b and of course as before you could just write it in this way so a less than x less than or equal to b okay so there's one more thing i want to just point out about um about these intervals i've taken i've taken a nice example where you you get something sensible if i look at the interval uh between the closed interval between zero and two then i'm going to be getting I get some numbers because there are numbers between 0 and 2. But if I get them the wrong way round, so if b is less than a, then a, b, well, it's all the stuff that is... bigger than a and less than b but b is less than a so there's nothing that is going to be less than b and also bigger than a so in that case you get the empty set so for example two zero or even the open interval two zero is empty similarly because there's if you put a equals b into a closed interval so if i take two comma two That's equal to the set 2, just containing 2, because it's all the elements between 2 and 2, but including the endpoints, so I get something. If I take all the elements that are strictly bigger than 2 and strictly less than 2, well, there's nothing.
So that's also equal to the empty set. Okay, there is one, actually. Tiny further piece of notation. Sometimes I want to include, I want to take all the numbers from all the way from minus infinity. We're not including infinity, but all the way from minus infinity up to a certain point.
So all the numbers all the way down from plus infinity to a certain point. So we also have the notation minus infinity comma. a and I can put either a Rounded bracket or a square bracket but this will be all the elements that are strictly less than a and I'm putting no limit on the on the On the left hand side I could have all the elements like that but including the end point So that would be x less than or equal to a. And similarly, I can go the other way.
I'll just give you one example. So if I have, let's call it b, why not? b up to infinity.
That's just going to be all the numbers that are bigger than b. So you can make all these sets in different ways. And then you can intersect them and you can see what you get. But anyway.
That will be all we need for this course.