Lesson Video: Cartesian Products Mathematics

Video Transcript

In this video, we will learn how to perform a Cartesian product and use the operations applied on sets. We will begin by explaining what we mean by a Cartesian product and discuss other notation we will use in this video.

The Cartesian product 𝐴 times 𝐵 of two sets 𝐴 and 𝐵 is the collection of all ordered pairs 𝑥, 𝑦, where 𝑥 is an element of 𝐴 and 𝑦 is an element of 𝐵. This means that the Cartesian product of two sets is itself a set, which consists of ordered pairs. A set of one of these ordered pairs is defined as a relation.

As an example, let’s consider the two sets, set 𝐴 containing the numbers two, three, and seven and set 𝐵 containing the elements five and seven. The Cartesian product 𝐴 times 𝐵 will therefore consist of ordered pairs. Each of these ordered pairs will have an 𝑥-value that is an element of set 𝐴 and a 𝑦-value that is an element of set 𝐵.

Let’s begin by considering the first element of set 𝐴, the number two. This can be paired with the element five in set 𝐵, giving us an ordered pair two, five. We can also have the ordered pair two, seven. We can then repeat this process with the three in set 𝐴, giving us the ordered pairs three, five and three, seven. Our final two ordered pairs are seven, five and seven, seven. There are six ordered pairs in the Cartesian product 𝐴 times 𝐵.

If, on the other hand, we looked at the Cartesian product 𝐵 times 𝐴, the 𝑥-values would be in set 𝐵 and the 𝑦-values in set 𝐴. We would have ordered pairs five, two; five, three; five, seven. We would also have the ordered pairs seven, two; seven, three; and seven, seven. Whilst 𝐵 times 𝐴 also has six ordered pairs, we can see that the Cartesian product 𝐴 times 𝐵 is not the same as 𝐵 times 𝐴. The only ordered pair that is the same is seven, seven. This is because the order matters. The 𝑥-coordinate has become the 𝑦 and vice versa.

We recall that the intersection of two sets 𝐴 and 𝐵 is denoted by 𝐴 n 𝐵. The intersection is the set of elements that is contained in both sets. In this question, 𝐴 intersection 𝐵 will be equal to the set containing the element seven as this is the only number that appears in set 𝐴 and set 𝐵. The union of sets 𝐴 and 𝐵 is denoted by 𝐴 u 𝐵. This is all the elements that are in set 𝐴 or in set 𝐵. We include all the elements in set 𝐴 plus any extra elements in set 𝐵 that are not contained in set 𝐴, in this case five. 𝐴 union 𝐵 is therefore equal to the set of two, three, seven, and five. As seven occurs in both set 𝐴 and set 𝐵, we do not need to write this twice.

We will now look at some questions where we need to calculate Cartesian products of sets with and without Venn diagrams.

Use the Venn diagram below to find the Cartesian product of 𝑋 and 𝑍.

The multiplication or times sign in the question means the Cartesian product. This is a collection of all ordered pairs. We can begin by writing the elements of set 𝑋, 𝑌, and 𝑍. Set 𝑋 just contains one element, the number six. Set 𝑌 contains the elements four and nine. Set 𝑍 contains the elements five, three, and four. Five and three appear in just set 𝑍, whereas four appears in the intersection of set 𝑍 and set 𝑌. We’re interested in the Cartesian product of set 𝑋 and set 𝑍.

As the first letter in our Cartesian product is 𝑋, the elements in this set will be the first number in our ordered pairs. The elements of set 𝑍 will be the second number. Our first ordered pair is therefore six, five. Next, we have six, three, the six from set 𝑋 and the three from set 𝑍. Finally, we have the ordered pair six, four. Each of the elements in set 𝑍 has now been matched with the six in set 𝑋. The Cartesian product of set 𝑋 and set 𝑍 is six, five; six, three; and six, four.

In our next question, we will not have a Venn diagram.

If set 𝑋 contains the elements eight, four, six; set 𝑌 is equal to six, seven; and set 𝑍 is equal to seven, find the Cartesian product of set 𝑋 and the intersection of set 𝑌 and 𝑍.

We recall that the Cartesian product of two sets is the collection of all ordered pairs. The intersection of two sets is those elements that occur in both sets, in this case in set 𝑌 and 𝑍. Set 𝑌 contains the elements six and seven, whereas set 𝑍 only contains the number seven. This means that the only number that appears in both sets is seven. The intersection of set 𝑌 and 𝑍 is equal to seven. We are told in the question that set 𝑋 contains the numbers eight, four, and six.

We now need to find the Cartesian product of these two sets. As our first value in the Cartesian product is 𝑋, each of the elements of 𝑋 will be the first number in the ordered pairs. Our first ordered pair is therefore eight, seven. Next, we have four, seven. Finally, we have the ordered pair six, seven. Each of our values in set 𝑋 has now been matched with the value of seven in the set 𝑌 intersection 𝑍. The Cartesian product contains the ordered pairs eight, seven; four, seven; and six, seven.

In our next question, we will introduce the union notation.

Determine the Cartesian product of 𝑍 minus 𝑌 and 𝑋 union 𝑌 using the Venn diagram below.

We recall that the Cartesian product is a collection of all ordered pairs. We also recall that the union of two sets is those elements that appear in either one of the sets. We can see from the Venn diagram that set 𝑋 contains just the number four; set 𝑌 contains the elements seven and nine; finally, set 𝑍 contains the elements three, eight, and seven.

The first part of our question is 𝑍 minus 𝑌. To perform this calculation, we begin with all the elements of 𝑍 and remove any elements that also appear in set 𝑌. The only element that appears in both sets is the number seven. This means that 𝑍 minus 𝑌 contains the elements three and eight.

As there was no intersection between set 𝑋 and set 𝑌 — that is, the circles in the Venn diagram do not overlap as there is no element in both sets — the union of 𝑋 and 𝑌 will be equal to all of the elements in 𝑋 and all of the elements in 𝑌. 𝑋 union 𝑌 is therefore equal to four, seven, nine. We can now find the Cartesian product of these two sets.

The first number of our ordered pairs will come from the set 𝑍 minus 𝑌. And the second number will come from the set 𝑋 union 𝑌. Our first pair is three, four. We then have three, seven and three, nine. We then repeat this with the eight, giving us eight, four; eight, seven; and eight, nine. The Cartesian product of 𝑍 minus 𝑌 and 𝑋 union 𝑌 contains the six ordered pairs three, four; three, seven; three, nine; eight, four; eight, seven; eight, nine.

In our next question, we’ll work backwards, as we are given the Cartesian product.

If the Cartesian product of set 𝑋 and set 𝑌 is equal to the set of ordered pairs eight, zero; eight, six; one, zero; one, six; three, zero; three, six, find set 𝑋.

The notation in the question denotes the Cartesian product. This is the set of all ordered pairs. As the first value of our Cartesian product is 𝑋, the first values in our ordered pairs must all be contained in set 𝑋. Likewise, the second values in the ordered pairs must be contained in set 𝑌. As there are three different 𝑋-values in our set of ordered pairs, set 𝑋 will contain these three values. They’re the numbers eight, one, and three. There are two different 𝑌-values, the numbers zero and six. As we are only interested in set 𝑋, the correct answer is eight, one, and three.

In our next question, we need to work out which of the relations our ordered pair would be an element of.

Given that 𝑋 is equal to the set zero, negative one and 𝑌 is equal to the set eight, negative four, negative three, negative two, then which of the following relations would negative one, negative four be an element of? Is it option (A) 𝑋 squared, (B) 𝑌 squared, (C) 𝑋 times 𝑌, or (D) 𝑌 times 𝑋?

The times notation in options (C) and (D) corresponds to the Cartesian product of sets 𝑋 and 𝑌 and sets 𝑌 and 𝑋, respectively. The Cartesian product is the set of all ordered pairs. Before starting this question, it is worth noting that option (A) 𝑋 squared is the same as the Cartesian product of set 𝑋 and set 𝑋. Likewise, 𝑌 squared is equal to the Cartesian product of set 𝑌 and set 𝑌.

The first value of our ordered pair, negative one, occurs in set 𝑋. The second value, negative four, occurs in set 𝑌. We can therefore conclude that option (C) the Cartesian product of 𝑋 and 𝑌 is the correct answer. This relation contains the ordered pair negative one, negative four.

An alternative method here would be to list all the ordered pairs of each relation. For example, the Cartesian product of 𝑋 and 𝑌 contains the ordered pairs zero, eight; zero, negative four; zero, negative three; zero, negative two; negative one, eight; negative one, negative four; negative one, negative three; and negative one, negative two. Once again, we see that negative one, negative four is contained in this Cartesian product.

In our final question, we will work out the union of two Cartesian products.

Determine the union of the Cartesian products 𝑋 and 𝑌 and 𝑌 and 𝑍 using the Venn diagram below.

We recall that the union of two sets is those elements that occur in the first set or the second set. The Cartesian product of any two sets is the set of all ordered pairs. We can see from the Venn diagram that set 𝑋 contains the element one; set 𝑌 contains the elements seven and five; set 𝑍 contains the elements two, zero, and five. Note that the number five appears in both set 𝑌 and set 𝑍 as it is the intersection of these two sets.

We can use the elements of set 𝑋 and set 𝑌 to work out the Cartesian product of 𝑋 and 𝑌. This contains the two ordered pairs one, seven and one, five. Note that the 𝑋-values are the first numbers in our ordered pair and the 𝑌-values are our second numbers. We can repeat this process for the Cartesian product of set 𝑌 and set 𝑍. In this Cartesian product, we have six ordered pairs: seven, two; seven, zero; seven, five; five, two; five, zero; and five, five.

We now need to work out the union of these two sets. This will include all the ordered pairs that are either in the Cartesian product of 𝑋 and 𝑌 or in the Cartesian product of 𝑌 and 𝑍. None of the ordered pairs are repeated. Therefore, we need to include all eight ordered pairs. The correct answer is the eight ordered pairs one, seven; one, five; seven, two; seven, zero; seven, five; five, two; five, zero; and five, five.

We will now summarize the key points from this video. We learnt in this video that the Cartesian product 𝐴 times 𝐵 of two sets 𝐴 and 𝐵 is the collection of all ordered pairs 𝑥, 𝑦, where 𝑥 is an element of 𝐴 and 𝑦 is an element of 𝐵. We also recalled that the intersection of two sets 𝐴 and 𝐵 is written 𝐴 n 𝐵. The intersection is all those elements that are contained in set 𝐴 and in set 𝐵. On a Venn diagram, this is the overlap of the two circles. The union of two sets 𝐴 and 𝐵 is written 𝐴 u 𝐵. This is the set of elements that appear in set 𝐴 or in set 𝐵. We only write the elements that appear in the intersection once when writing the union. We also saw that in some of our questions, a Venn diagram can be useful to help solve these type of problems.