Video Transcript
Express the set of real numbers ℝ
in the form of an interval.
In this question, we want to
represent the set of real numbers in interval notation. So, we can begin by recalling what
is meant by interval notation. First, we let 𝑎 and 𝑏 be real
numbers with 𝑎 less than 𝑏. We can then describe the set of all
real numbers between 𝑎 and 𝑏 by using interval notation. It is the open interval from 𝑎 to
𝑏. There are two common notations for
this set, either using reverse brackets or parentheses. In either case, the first number
gives us a lower bound for the numbers in the set, and the second number gives us an
upper bound for the numbers in the set.
We can also describe intervals
where the endpoints are included by using brackets. These are called closed
intervals. So, the closed interval from 𝑎 to
𝑏 is the set of all real values between 𝑎 and 𝑏, which includes both 𝑎 and
𝑏. If we try to represent the set of
real numbers using this notation, then we run into a problem. We know that there is no largest or
smallest real number. So, we cannot bound this set using
real numbers.
To get around this, we introduce
the symbols of positive and negative ∞ into interval notation. These symbols mean that there is no
bound on that side. For instance, the left-closed,
right-open interval from 𝑎 to ∞ will include all real values greater than or equal
to 𝑎.
There are two things worth
highlighting about this notation. First, we use ∞. Since we are bounding above, we
want a positive number. Second, we must use an open
interval on the unbounded side, since ∞ is not included in the set. In the same way, we use negative ∞
to show that there is no lower bound. So, the open interval from negative
∞ to 𝑏 means all of the real values less than 𝑏. If we combine these ideas, then the
open interval from negative ∞ to ∞ means all of the real values with no upper or
lower bound. That is just the set of real
numbers ℝ.
Hence, the open interval from
negative ∞ to ∞ is the set of all real numbers in interval notation.
