Lesson Video: Sets and Venn Diagrams Mathematics

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.