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.