Question Video: Finding the Complement of an Interval Mathematics

Given that 𝑥 (−∞,9], find 𝑥′.


Video Transcript

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 number.

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 line.

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 ∞.

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