Video Transcript
In this lesson, we will learn how
to determine limited and unlimited intervals. An interval is a way of describing
a subset of the real numbers. And we do that between an upper and
lower bound.
For example, we can describe the
set of positive numbers 𝑥 which are less than two, as the inequality shown. The equivalent set in interval
notation is represented using these round parentheses or brackets. So if we write 𝑥 is an element of
the open interval from zero to two, it means it’s between zero and two, not
including zero and two. We call this a limited interval,
since it has both an upper and a lower bound.
So let’s generalize this. Suppose we have two real numbers 𝑎
and 𝑏, where 𝑏 is bigger than 𝑎. Using the round brackets means we
have an open interval from 𝑎 to 𝑏. And that’s the interval between 𝑎
and 𝑏, but not including 𝑎 and 𝑏. What would we do then if we wanted
to represent the following inequality as an interval? This inequality tells us that 𝑥 is
between zero and two and could be equal to zero and two. Well, this time, we use the square
brackets. And we read this as 𝑥 is an
element of the closed interval from zero to two. If we generalize this, the closed
interval from 𝑎 to 𝑏 is the interval between 𝑎 and 𝑏, including 𝑎 and 𝑏. It’s worth noting that we can mix
and match these brackets to represent different types of intervals. Let’s demonstrate these.
We’ve shown that for real numbers
𝑎 and 𝑏, using the round brackets gives us the open interval from 𝑎 to 𝑏. And this is equivalent to saying
the set containing 𝑥 where 𝑥 is greater than 𝑎, less than 𝑏, and a real
number. Then, if we use square brackets,
that’s the closed interval from 𝑎 to 𝑏. And in this case, 𝑥 can be greater
than or equal to 𝑎 and less than or equal to 𝑏. If we use one round bracket and one
square bracket as shown, we can read this as the left-open right-closed interval
from 𝑎 to 𝑏. And this tells us that we are
interested in the set containing 𝑥 where 𝑥 is greater than 𝑎, less than or equal
to 𝑏, and still a real number. And of course, we can reverse these
brackets to give us the left-closed right-open interval.
Now these definitions do allow for
a few interesting cases. Consider, for instance, the closed
interval from 𝑏 to 𝑏. Well, that contains all real
numbers which satisfy the inequality 𝑥 is greater than or equal to 𝑏 and less than
or equal to 𝑏. But that’s just the number 𝑏. So that interval is equivalent to
the set containing the single number 𝑏. In a similar way, the open interval
from 𝑎 to 𝑎 contains all real numbers, which satisfy the inequality 𝑥 is greater
than 𝑎 and less than 𝑎. Now, of course, there are no
numbers which are both greater than and less than 𝑎. So this involves the empty set. Finally, if 𝑏 is less than 𝑎,
then all of these intervals are empty.
Let’s now demonstrate how to
describe a given interval notation using number lines.
Which of the following figures
represents the interval left open, right closed from negative one to zero?
Now, when we read this
interval, we read it as left open, right closed. So if we say 𝑥 is an element
of this interval, then we’re saying 𝑥 can be greater than negative one. But it can also be less than or
equal to zero. Now, if we look at all of the
intervals represented on number lines, we do see that they do lie between
negative one and zero. In order to narrow this down
further then, let’s remind ourselves what the dots or circles at the end of each
interval mean. If we have an open or empty
dot, that means greater than or less than. Then, if we have a solid or
closed dot, that means we need to include the number that this lies above. So it means less than or equal
to or greater than or equal to.
In order to represent the
left-open right-closed interval from negative one to zero then, we need to have
an open dot at negative one and a closed dot at zero. Looking at each of the number
lines, we see that’s option (b). In fact, let’s go through and
represent the intervals for the remaining three number lines.
In example (a), we have an open
dot at each end. This means we have an open
interval from negative one to zero. Then, in option (c), we’re
looking at the opposite to option (b). It’s left closed, right
open. Finally in option (d), both of
our dots are closed. So we need to include both
negative one and zero in our interval. It’s the closed interval from
negative one to zero.
We’ve now demonstrated how to
represent intervals on number lines. In our next example, we’re going to
consider how to use the definition of this interval notation to determine whether a
number is an element of some given interval.
Which of the following is
true? Is it (A) five is an element of
the open interval from root six to root 26? Or is it (B) five is not an
element of the open interval from root six to root 26?
When we read these two options,
we read them, respectively, as five is an element and five is not an element of
that interval. In other words, since the
interval is open, in option (A), we’re saying five is between root six and root
26. And in option (B), we’re saying
it’s not between these two numbers. In order to identify which is
true, we could use a calculator to evaluate root six and root 26.
We’re going to use a
non-calculator method though. This method involves quoting
the first six square numbers. One squared is one, two squared
is four, all the way up to six squared is 36. Then, we compare these to the
first number in each interval. Since taking the square root is
the inverse operation to squaring a number, we can say that the square root of
six must be between the square root of four and the square root of nine. In other words, the square root
of six is between two and three. Then, we look at the second
number in our interval, the square root of 26. The square root of 26 must be
between the square root of 25 and the square root of 36. In other words, the square root
of 26 is greater than five and less than six.
Now, we have indeed shown that
five is less than the square root of 26. And of course, since five is
bigger than three, by definition, five must also be bigger than the square root
of six. If we represent this in
inequality notation, we can say that five is greater than root six and less than
root 26 and that in turn means we can use interval notation to say that five is
an element of the open interval from root six to root 26. The correct answer is (A).
Now, up until this point, we’ve
looked at limited intervals only, in other words, intervals which had both an upper
and a lower bound. Let’s now consider what we mean
when we talk about an unlimited interval.
We say that an interval is
unlimited or unbounded if it does not have both upper and lower bounds. So for instance, we wanted to
represent the inequality 𝑥 is greater than or equal to 𝑎 using interval
notation. This is an unlimited interval. It has a lower bound only. It tells us that 𝑥 can be greater
than or equal to 𝑎. But that could go all the way to
positive ∞. Since we can’t easily define ∞ as a
real number, we have to use a round bracket to represent that end of the
interval. So, if 𝑥 is greater than or equal
to 𝑎, this is equivalent to saying 𝑥 is an element of the left-closed right-open
interval from 𝑎 to ∞. And of course, we can use a round
bracket to represent a strict inequality. So 𝑥 is less than 𝑏 would be
equivalent to saying 𝑥 is an element of the open interval from negative ∞ to
𝑏.
With this in mind, let’s look at
how to represent a given set as an unlimited interval.
Express the following set using
interval notation. 𝑋 is equal to the set
containing lowercase 𝑥 where 𝑥 is greater than or equal to two and 𝑥 is a
real number.
First, we know that since 𝑥
could be greater than or equal to two but has no upper bound, we’re going to
represent this as an unlimited interval. It has no upper limit. So we can write the upper limit
as positive ∞. Remember, of course, that when
we do so, we cannot easily define or quantify positive ∞. And so, we use a round bracket
to represent an open interval at that point. We do, however, want to include
the number two. And so we represent our answer
as a left-closed right-open interval. Specifically, the set 𝑋 using
interval notation is represented by the left-closed right-open interval from two
to ∞.
Now that we’ve demonstrated how to
represent an unlimited set using interval notation, let’s determine which of a group
of statements about a number is true.
Which of the following is
true? Is it (A) root 49 is an element
of the open interval five to ∞? Is it (B) root 49 is not an
element of this open interval? Is it (C) root 49 is a subset
of this interval? Or (D) root 49 is not a subset
of this interval.
Now, when we read options (C)
and (D), we said that this means it’s a subset or not a subset of this interval,
respectively. Now, of course, for this
notation to be valid, both root 49 and the open interval from five to ∞ must be
sets. But of course, the square root
of 49 is not a set. And this means we can instantly
disregard options (C) and (D).
In order to determine which of
(A) and (B) is true, let’s look to evaluate the square root of 49. Now, this is one that we should
know by heart. The square root of 49 is equal
to seven. Then, we can note that the
lower bound of each of our intervals is five. And we know that seven is
greater than five. This in turn means that the
square root of 49 must be greater than five. But of course, it’s only
seven. It’s definitely not as big as
∞. So we can say that the square
root of 49 is greater than five and less than ∞. And that’s equivalent to saying
that the square root of 49 is an element of the open interval from five to
∞. So the correct answer is
(A).
Now, since intervals are sets, we
can actually perform set operations on these intervals. So we can take the union of
intervals to represent real numbers in either interval. We can find the intersection of
intervals. That’s the real numbers in
both. We can find the difference between
two intervals, which removes elements of one interval from another. And finally, we can take the
complement of an interval. And that represents all of the real
numbers not in that interval. Let’s demonstrate this with an
example.
Which of the following
expressions represents the set shown on the number line? Is it (A) the set of real
numbers minus the open interval from negative four to one? (B) The set of real numbers
minus the closed interval from negative four to one. Is it (C) the set of real
numbers minus the left-closed right-open interval from negative four to one? Or (D) the set of real numbers
minus the left-open right-closed interval from negative four to one. Or is it (E) the union of the
open interval from negative ∞ to negative four and the left-closed right-open
interval from one to ∞?
Well, in fact, there are a
number of ways to answer this problem. Let’s begin by identifying the
regions that we are interested in on our diagram. Suppose we say that 𝑥 is in
the set shown on the number line. We can say that 𝑥 must be less
than or equal to negative four since this dot is solid. We can also say that 𝑥 must be
greater than one since this dot is open. Now, if we represent each of
these individually using interval notation, that’s the left-open right-closed
interval from negative ∞ to four and the open interval from one to ∞.
To show that 𝑥 could be in
either of these intervals, we use the union notation. So, one way we have to
represent this is to say that it’s the union of the left-closed right-open
interval from negative ∞ to four and the open interval from one to ∞. And of course, that allows us
to disregard option (E) since the brackets are the wrong way round.
Let’s now consider what this
set is not. Since 𝑥 can be less than or
equal to negative four and greater than one, it cannot also be greater than
negative four and less than or equal to one. In other words, it’s not in the
left-open right-closed interval from negative four to one, which we can
represent using the complement notation as shown. But of course, since this is
equivalent to saying it’s a set of real numbers excluding this set, we can
represent this equivalently as the set of real numbers minus the left-open
right-closed interval from negative four to one. And that’s option (D).
Now, we can actually generalize
what we’ve just shown. In other words, if 𝐼 is some
interval, then the complement of 𝐼 is simply the set of real numbers minus that
interval 𝐼. Now, of course, this is true for
any subset of ℝ, not just intervals. But it’s useful to commit this
result to memory.
Let’s now recap the key points from
this lesson.
In this lesson, we saw that for
real numbers 𝑎 and 𝑏, we can define intervals using round and square brackets. We saw that we can mix and match
these brackets, giving us half-open or half-closed intervals. We also saw that an interval is
called unlimited if it does not have both an upper and lower bound. And we can use negative ∞ and
positive ∞ with a round bracket to represent these. We saw that we can use set
operations, such as union, intersection, difference, and complement on intervals,
and that if we have some interval 𝐼, then the complement of 𝐼 can be equivalently
written as the set of real numbers minus 𝐼.
