### Video Transcript

In this video, we will learn how to
represent a set on a Venn diagram. We will begin by defining set
notation and looking at what a Venn diagram looks like. Set notation is used in mathematics
to list numbers, outcomes, or objects. It uses curly brackets, which is
sometimes referred to as braces. Objects placed within the brackets
are called the elements of a set. For example, the set of
single-digit even numbers would be written as follows. 𝐸 is equal to two, four, six, and
eight. These elements — in this case, the
numbers two, four, six, and eight — can be written in any order. We can use any capital letter to
name the set. In this case, we have used 𝐸 for
even.

We will now at what a Venn diagram
looks like. Venn diagrams are very useful
constructs made up of circles that sometimes overlap. Whilst they appear in different
areas of mathematics, they are most common when dealing with sets and
probability. In this video, we will focus purely
on the questions involving sets.

If we consider the Venn diagram
drawn, we have a circle labeled 𝑋 inside a rectangle labeled 𝑍. As all five numbers are inside the
rectangle, we can say that the set 𝑍 contains the numbers one, two, three, four,
and five. Set 𝑋 will contain all the numbers
inside the circle, in this case, two, three, and five. These are actually all the prime
numbers between one and five. Venn diagrams are often set up to
show practical or real-life problems like this. Whilst we won’t consider problems
of this type in this video, the set of numbers 𝑋 prime is equal to one and
four. 𝑋 prime denotes those numbers that
are not in set 𝑋. As one and four are outside of
circle 𝑋, they are in set 𝑋 prime. These are the numbers between one
and five that are not prime numbers.

Let’s now consider what happens
when we have overlapping circles. In this example, we have two
circles, set 𝑋 and set 𝑌, inside the rectangle 𝑍. Once again, 𝑍 will be all the
numbers contained in the Venn diagram, in this case, the integers or whole numbers
from one to nine. Set 𝑋 contains the numbers two,
four, six, and eight. These are all the numbers inside
circle 𝑋 and other multiples of two or even numbers. The numbers contained in set 𝑌 are
three, six, and nine. These are the multiples of
three.

We notice that the number six
appears in set 𝑋 and in set 𝑌. This is because the six is in the
intersection of the circles 𝑋 and 𝑌. We denote intersection by a
lowercase n. The set of numbers in the
intersection 𝑋 and 𝑌 is therefore equal to six. In this particular case, this is
because six is the only number between one and nine that is a multiple of two and a
multiple of three. We will now look at some specific
questions involving sets in Venn diagrams.

Write 𝑌 using the listing
method.

The listing numbers involve writing
a list of numbers using set notation. In this case, we’re looking at the
set 𝑌. The numbers in set 𝑌, known as the
elements, are written inside curly brackets. In this case, we have six
numbers. In ascending order, these are one,
three, four, six, seven, and eight. When writing a set, however, the
order does not matter. For example, 𝑌 is equal to the set
of numbers eight, one, three, six, four, and seven. Any order that contains these six
elements or numbers is correct.

Our next question involves a Venn
diagram with two circles.

Write 𝑋 using the listing
method.

In this question, we have two
intersecting circles, circle 𝑋 and circle 𝑌. We could, therefore, write the set
of elements or numbers in circle 𝑋 and the set of numbers in circle 𝑌. In this question, we’re interested
in 𝑋. We write the numbers contained in a
set using curly brackets. There are five numbers in total
contained in set 𝑥. The numbers nine, six, and three
are only in set 𝑋. The numbers eight and five are in
the intersection. Therefore, they’re in set 𝑋 and in
set 𝑌. Set 𝑋 contains the numbers nine,
six, three, eight, and five.

These numbers can be written in any
order. It is often useful, however, to
write them in ascending or descending order, for example, three, five, six, eight,
and nine. Whilst we’re not asked to work it
out in this case, set 𝑌 contains the numbers one, five, seven, and eight. One and seven only appear in set
𝑌. And once again, eight and five are
in the intersection. This is the overlap between both
circles.

Our next question asks us to
identify the correct Venn diagram that corresponds to two sets.

If 𝑋 equals the set of numbers
seven, three, nine, one and 𝑌 equals the set of numbers nine, seven, two, which
Venn diagram represents the two sets?

In this question, we could begin by
looking at each of our Venn diagrams and list the set 𝑋 and the set 𝑌. In this question, however, this
might not help us that much. In each of (A), (B), and (C), the
numbers one, three, seven, and nine exist in set 𝑋. This is because all four of those
numbers are inside circle 𝑋. Likewise, circle 𝑌 in all three of
these options contain the numbers two, seven, and nine. This seems to suggest that all
three of these options satisfy the conditions. The key word in this question,
however, is the word “and.”

This means that we need to focus on
the intersection or overlap of the two circles denoted by 𝑋 n 𝑌. The lowercase symbol that looks a
bit like an n means the intersection. The numbers seven and nine appear
in both sets. They’re in set 𝑋 and in set
𝑌. This means that these numbers need
to appear in the intersection on our Venn diagram. In option (B), there is no
intersection between the two circles. Therefore, this cannot be the
correct answer. In option (C), the only number that
intersects circle 𝑋 and circle 𝑌 is seven, so this is also incorrect. In option (A), we see that seven
and nine appear in the intersection.

The correct answer is option (A) as
the number seven, three, nine, and one are in set 𝑋, nine, seven, and two are in
set 𝑌, and the number seven and nine appear in the intersection. We could have immediately ruled out
option (D) as set 𝑋 here contains the numbers one, two, three, seven, and nine. Set 𝑌 contains seven and nine. While seven and nine do appear in
both sets, the individual sets are incorrect.

We will now look at a question
where one set is a subset of another.

Using the given Venn diagram, list
the elements of set 𝑋.

In this diagram, we have a circle
𝑌 inside circle 𝑋, which in turn is inside the square 𝑍. As the circle 𝑌 is contained
entirely inside circle 𝑋, we can say that 𝑌 is a subset of 𝑋. This means that all the values in
set 𝑌 will also be in set 𝑋. Set 𝑌 can be written using curly
brackets as shown. It contains the numbers or elements
three, five, and six. These numbers can be written in any
order. In this question, we’re interested
in those elements in set 𝑋.

As 𝑌 is a subset of 𝑋, 𝑋 must
include all the numbers in set 𝑌, three, five, and six. We also have the extra numbers one
and nine that are inside circle 𝑋 but not in circle 𝑌. The elements in set 𝑋 are three,
five, six, one, and nine. Once again, these could be written
in any order. Whilst we’re not asked to in this
question, we could also write set 𝑍. This would contain all the elements
inside 𝑋, three, five, six, one, and nine together with the number two, which is
outside set 𝑋, but inside set 𝑍.

Our final question requires us to
identify the correct Venn diagram that represents the three sets.

If 𝑋 equals the set of numbers
six, zero, three, seven, eight; 𝑌 is equal to the set of numbers eight, three,
five, two; and 𝑍 equals eight, zero, one, four, five, which Venn diagram represents
the sets?

A starting point in this question
might be to consider set 𝑋 and see which of our diagrams contain the numbers six,
zero, three, seven, and eight. Unfortunately, this alone does not
help as all four of our options have these five numbers. Set 𝑌 contains the numbers eight,
three, five, and two. Once again, all four circles
contain these four numbers. We know that set 𝑍 contains the
numbers eight, zero, one, four, and five. Options (A), (C), and (D) all
contain these five numbers in circle 𝑍. This means that we can only
eliminate option (B) using this start point.

In questions such as this, it is
far more important to look at the intersections or overlaps between the circles. The intersection notation is
written as shown. Let’s first consider those numbers
that appear in set 𝑋, set 𝑌, and set 𝑍. The only number that appears in all
three sets is eight. This means that we must have an
eight in the intersection of all three circles. Once again, options (A), (C), and
(D) all satisfy this. Let’s now consider which numbers
appear in set 𝑋 and 𝑌, 𝑋 and 𝑍, and 𝑌 and 𝑍. The only number apart from eight
that appears in set 𝑋 and set 𝑌 is three. This means that the set of numbers
in the intersection 𝑋 and 𝑌 is eight and three. The number three must appear in the
overlap between set 𝑋 and set 𝑌.

In options (C) and (D), this is
true. However, in set (A) there is no
number in this intersection, so we can rule out this option. The number zero exists in set 𝑋
and in set 𝑍. This means that the intersection of
𝑋 and 𝑍 contains eight and zero. We must have a zero in the overlap
between 𝑋 and 𝑍. This is true for option (C), but
not for option (D) as there is no number in the intersection between 𝑋 and 𝑍
only. This suggests that option (C) is
the correct answer.

We will check this by considering
the intersection of 𝑌 and 𝑍. The number five appears in set 𝑌
and set 𝑍. It also appears on the Venn diagram
in the intersection of these two circles. The intersection of set 𝑌 and set
𝑍 needs to contain eight and five, which is correct. The Venn diagram that represents
the three sets given is option (C). In this type of question, the most
important part is to look at the intersection of the different sets.

We will now summarize the key
points from this video. We saw in this video that a set of
numbers can be written as shown. We use a capital letter to define
any set. We use curly brackets, sometimes
known as braces, at either end. The numbers inside the curly
brackets are known as the elements of the set. We also saw in this video that a
Venn diagram can be used to identify sets. We can match a given set or sets
with a Venn diagram, or vice versa.

In the diagram shown, we have sets
𝑋, 𝑌, and 𝑍. Set 𝑋 contains the numbers three
and one. These can be written in any
order. Set 𝑌 contains the elements or
numbers two, four, and one. All of the numbers three, one,
four, and two, together with the number seven, are in set 𝑍. We found that this means that set
𝑋 and set 𝑌 are subsets of set 𝑍. All of their elements are contained
in set 𝑍.

We also saw the importance of the
intersection or overlap of our two sets. In this example, the number one
appears in set 𝑋 and in set 𝑌. This means that it must appear in
the overlap of the two circles. And the set of numbers in the
intersection of 𝑋 and 𝑌 is one. The types of questions that we have
seen in this video can help us go on and solve problems involving probability.