### 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.