Lesson Video: Expressing a Set Mathematics

In this video, we will learn how to list sets of information.


Video Transcript

So in this lesson, what we’re gonna be looking at is expressing a set. What this means is actually listing values that could be in a particular set of parameters. And each of these values is known as an element. So even though we say values, as I said, we should say elements because they don’t have to be numbers, for example, the set of vowels a, e, i, o, u. But then we can have numbers like the set of odd numbers, which is one, three, five, et cetera. So we now know a little bit about what expressing a set is.

Now what we’re gonna do is take a look at some questions that will show us how we would express particular sets. So ready, set, go. Let’s have a look at the first question.

Using the listing method, express the set of the days of the week.

So what this question means by the listing method is listing out each of the elements of the set that we’re looking at. It is important to remember to use the correct terminology. So if we’re talking about a part of our set — a part of our set is called an element. And we can already say that we know that our set is gonna have seven elements because if we’re talking about the days of the week, we know that there are seven days of the week. And whenever we’re about to list a set, what we use is some set notation to represent that we’re gonna be listing a set. And that is the curly bracket that we have here. We’re gonna have one at each end of our set.

So our first element is gonna be Saturday. You could start with any day. I’ve chosen to start with Saturday, then Sunday, Monday, Tuesday. Then next, we have Wednesday and then Thursday and, finally, Friday. And we can quickly double check to make sure that we’ve got the seven elements we expected, and we have. So therefore, we can say that the set of the days of the week is Saturday, Sunday, Monday, Tuesday, Wednesday, Thursday, and Friday.

Okay, great. So what we’ve done is we’ve expressed a set and we’ve shown how to do that using the listing method. And we used a little bit of notation with our curly brackets. And we’ve also talked about the elements being each part of our set. So now what we’re gonna do is take a look at an example which includes numerical values and see how we’d list those.

𝑌 is the set of digits in the number 90,590. Write 𝑌 using the listing method.

Well, the first thing we need to do in this question, to enable us to write 𝑌 using the listing method, is identify the digits within our number. Well, first of all, we can see that the digit nine appears twice. Then we have the digit zero, which also appears twice, and finally the digit five. Now, when we’re gonna list these digits, so we’re gonna list the set of digits, which is 𝑌, we only need to list each digit once. So therefore, the set 𝑌 would equal nine, five, and zero. And it’s worth noting here that a common mistake would be to list all of the digits that we have, for instance, two nines, two zeros, and a five. So, It’s also worth noting that it doesn’t matter which order we put them in. I’ve just put them here in descending order.

So we’ve looked at a couple of examples now, one that involved nonnumerical values, one that involves numerical values. What would we take a look at next? Well, next, we’re gonna have a look at what we do if we’ve got a very big set or, in fact, an infinite set.

𝑋 is the set of odd numbers greater than eight. Write 𝑋 using the listing method.

Well, in this question, if we want to list our set and find each of the elements, then what we’re gonna need to do is take a look at these two key bits of information. We want the set to be odd numbers, but they must be greater than eight. Well, we’ve got one problem. The odd numbers greater than eight is gonna be a lot of numbers. In fact, it’s gonna be ∞ numbers. So our set is gonna have ∞ elements because it’ll keep going on and on and on. So what are we gonna do?

Well, first of all, what we’re gonna do is list our first value because the first odd number that’s greater than eight is nine. And then, what we’re gonna do is list a couple more values, 11 and 13. But then, instead of having to list lots and lots and lots of different values or lots of different elements, all we do is we put three little dots. And this means continued, because, as we’ve said, there’ll be infinite number of different elements within this set. So we can say that if 𝑋 is a set of odd numbers, then 𝑋 can be written as nine, 11, 13, et cetera. And I’ve put that inside of our curly brackets, which are part of our set notation and tells us that it represents a set of values.

So we’ve looked at listing different sets and we’ve shown how we can do this using different notation. So finally, what we’re gonna do is we’re just going to show you how you’d list elements on their own. So not entire sets, but just still elements of a set.

Write the elements of the set, the odd numbers between, but not including, 799 and 805.

So the keyword here in this question is element because what it means is we want to write the individual parts of our set. And these are known as elements. We can also see the next bit of useful information. And that is that we’re looking at odd numbers. And it must be between, but not including, 799 and 805. So our first value will not be 799; it will be 801 because this is the next odd number. Then, the next odd number that meets our criteria is 803 and then, finally, 805. This will also not be included because we’re told in the question that 799 and 805 are not included. So therefore, the elements of the set, the odd numbers between, but not including, 799 and 805 are 801 and 803. There are two elements.

So we’ve now come to the end of the lesson because we’ve shown how we can list sets. And we’ve listed sets that include numerical and nonnumerical values. We’ve also looked at elements and how we can list individual elements. And we’ve shown how you’d list sets that have infinite number of values.

So now what we’re gonna do is take a look at the key points. The first key point we’ve got is that an element is a value or part of a set. So any individual part of our set is called an element. We’ve also seen that we use set notation. So we have these curly brackets that show that something is in fact a set. And we’ve also shown that if we have an infinite number of elements in a particular set, then we can show this by listing the first two or three elements, then having these three dots after, which says that it’s gonna carry on infinitely. And what we’ve shown is with our examples that a set can be numerical or nonnumerical. And any set can have any number of parameters, which help us decide upon what that set is gonna be and what the elements within it are.

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