Given that 𝑥 is equal to the
left-open, right-closed interval from negative ∞ to nine, find 𝑥 prime.
In this question, we are given a
set 𝑥 in interval notation and asked to use this to find the set 𝑥 prime. To find this set, we can begin by
recalling that interval notation gives us the bounds of the values that are included
in the set. In this case, we note that the
lower bound is negative ∞ and the upper bound is nine and the set is closed at
nine. So we want to include the value of
nine in the set 𝑥. Therefore, 𝑥 is the set of real
values greater than negative ∞ but less than or equal to nine. For simplicity, when using
set-builder notation, we often leave out any bounds at positive or negative ∞ since
they are not needed.
We now need to recall what is meant
by the notation 𝑥 prime. We can recall that the prime
notation for sets means the complement of the set. Therefore, we say that 𝑎 is a
member of the complement of 𝑥 if it is not a member of 𝑥 and it is a real
This then gives us two ways of
answering the question. The first method is to note that
the complement of 𝑥 will include all real numbers that are not less than or equal
to nine. This is the same as saying that its
elements are greater than nine. We can write this in interval
notation as the open interval from nine to ∞. We can use either parentheses or
reverse brackets to show that the interval is open at the endpoints.
While this method works, it is
quite difficult, particularly if 𝑥 is a complicated set or if the set operations
are complicated. So we often need to visualize these
sets to help us. We can do this by using a number
First, we sketch 𝑥 on the number
line. We know that 𝑥 is all of the real
numbers less than or equal to nine. So we sketch a solid circle at nine
to show that this value is included in 𝑥. Then, we draw an arrow to the left
to show that all values less than nine are elements of 𝑥. We then want to find the complement
of 𝑥 on the number line, that is, all of the real values not in 𝑥. We can start by noting that nine is
not included in the complement of 𝑥. So we sketch a hollow circle at
nine. We then note that we need to
include all of the real values above nine in the complement of 𝑥. So we can sketch this set by adding
an arrow to the right of nine as shown.
Once again, we see that the
complement of 𝑥 includes all real values greater than nine. So it is the open interval from
nine to ∞.