Question Video: Finding the Intersection of Two Intervals | Nagwa Question Video: Finding the Intersection of Two Intervals | Nagwa

Question Video: Finding the Intersection of Two Intervals Mathematics • Second Year of Preparatory School

Given that 𝑥 = [−8, 3) and 𝑦 = [−4, −3) find 𝑥 ∩ 𝑦.

04:30

Video Transcript

Given that 𝑥 is equal to the left-closed, right-open interval from negative eight to three and 𝑦 is equal to the left-closed, right-open interval from negative four to negative three, find the intersection between 𝑥 and 𝑦.

In this question, we are given two sets 𝑥 and 𝑦 in interval notation. We need to use this to find the intersection between the two sets. To do this, let’s start by recalling what we mean by the intersection of two sets.

We can recall that we say that 𝑎 is a member of the intersection between 𝑥 and 𝑦 if 𝑎 is a member of 𝑥 and 𝑎 is a member of 𝑦. In other words, the intersection between 𝑥 and 𝑦 includes all of the elements in both sets. To find the intersection between 𝑥 and 𝑦, let’s recall what is meant by the interval notation for 𝑥 and 𝑦.

Let’s start with 𝑥. In interval notation, the first number is a lower bound of the set and the second number is an upper bound of the set. We also note that we have a bracket at negative eight and a parenthesis at three. This means that we want to include negative eight but not include three. So 𝑥 is the set of all real numbers between negative eight and three, where we include negative eight. In the same way, we can see that 𝑦 is the set of real values between negative four and negative three, where we include negative four.

We now want to find the intersection of these sets, that is, all of the values in both sets. To do this, it is a good idea to represent these sets visually on a number line. Let’s start with 𝑥. We know that 𝑥 contains all of the real values between negative eight and three. However, we need to include negative eight and not three. We can show this on a number line by drawing a solid circle at negative eight to show that it is included and a hollow circle at three to show that it is not included in this set. We then connect these with a line to show that all of the values between the endpoints are in the set.

We can follow the same process for 𝑦. We see that 𝑦 includes negative four but does not include negative three. So we sketch a solid circle at negative four and a hollow circle at negative three and connect them with a line to represent the set 𝑦. The intersection of these two sets includes every value in both of the sets. We can see in the diagram that the lowest value in both sets is negative four. We can then see that all of the values up to negative three are elements of both sets. So they are in the intersection.

Importantly, we can note that negative three is not a member of 𝑦. So it is not a member of the intersection. There are no other elements in the intersection. So the intersection is the left-closed, right-open interval from negative four to negative three. We can represent this in interval notation in two different ways: either using a parenthesis to show that the interval is open on the right or by using a backwards bracket.

It is also worth noting that in our diagram, we showed that every element of 𝑦 is an element of 𝑥. So 𝑦 is a subset of 𝑥. This means that 𝑥 intersect 𝑦 will just be equal to 𝑦. We can see this in our answer. The intersection between 𝑥 and 𝑦 is equal to 𝑦, which is the left-closed, right-open interval from negative four to negative three.

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