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Lesson Explainer: Intervals Mathematics

In this explainer, we will learn how to determine limited and unlimited intervals.

An interval is a way of describing a subset of the real numbers between two given values. For example, we can describe the set of positive numbers, π‘₯, that are less than 2 as an inequality: 0<π‘₯<2. The equivalent set in interval notation is ]0,2[, so saying π‘₯∈]0,2[ means that it is between 0 and 2. These represent the same set of numbers: those above 0 and less than 2. We call this a limited or bounded set since all of the values in the set are between two numbers, and we call the values of 0 and 2 the endpoints of this interval.

We can also represent this set by using a number line; we want to include all of the values between 0 and 2 but not the endpoints. We do this by using open circles to show that 0 and 2 are not included, as shown.

Before we formally define interval notation, we should note that sometimes we want to include the endpoints in our set. To do this, we introduce the notation [0,2], which is equivalent to the inequality 0≀π‘₯≀2. In other words, it is the set of all values greater than or equal to 0 and less than or equal to 2. When both endpoints are included in the interval, we call it a closed interval; if neither endpoint is included, then we say it is an open interval.

We can also represent this interval on a number line. This time, we used closed circles to represent that the endpoints are included in the set as shown.

It is worth noting that we can include a single endpoint and exclude the other by using each type of parenthesis. For example, the interval ]0,2] would be equivalent to the inequality 0<π‘₯≀2 and we could represent this interval on a number line as shown.

We are now ready to define interval notation formally, as follows.

Definition: Limited Intervals

For real numbers π‘Ž and 𝑏, where π‘Ž<𝑏, we can define the following limited intervals:

  • ]π‘Ž,𝑏[={π‘₯βˆΆπ‘Ž<π‘₯<𝑏,π‘₯βˆˆβ„}: this is called the open interval from π‘Ž to 𝑏. This interval contains all of the values between π‘Ž and 𝑏 but does not include π‘Ž or 𝑏.
  • [π‘Ž,𝑏]={π‘₯βˆΆπ‘Žβ‰€π‘₯≀𝑏,π‘₯βˆˆβ„}: this is called the closed interval from π‘Ž to 𝑏. This interval contains all of the values between π‘Ž and 𝑏 including both π‘Ž or 𝑏.

We can also define the following two half-open (or half-closed) intervals:

  • ]π‘Ž,𝑏]={π‘₯βˆΆπ‘Ž<π‘₯≀𝑏,π‘₯βˆˆβ„}: this interval contains all of the values between π‘Ž and 𝑏 but does not include π‘Ž; however, it does contain 𝑏.
  • [π‘Ž,𝑏[={π‘₯βˆΆπ‘Žβ‰€π‘₯<𝑏,π‘₯βˆˆβ„}: this interval contains all of the values between π‘Ž and 𝑏 but does not include 𝑏; however, it does contain π‘Ž.

This definition allows for a few interesting cases.

First, we can consider the interval [𝑏,𝑏]; this set would include all real numbers that satisfy the inequality 𝑏≀π‘₯≀𝑏, which is just the number 𝑏. Hence, [𝑏,𝑏]={𝑏}.

Second, we can consider the interval ]π‘Ž,π‘Ž[; this would be all real numbers that satisfy the inequality π‘Ž<π‘₯<π‘Ž. Since there are no numbers that are both greater than and less than π‘Ž, this set has no members. Hence, ]π‘Ž,π‘Ž[=βˆ….

Third, if 𝑏<π‘Ž, then these intervals are all empty. For example, the interval ]2,1[ would include all real numbers greater than 2 that are also less than 1, which is not possible, so the set is empty.

Let’s now see an example of describing a given interval notation on a number line.

Example 1: Identifying the Number Line That Represents a Given Interval

Which of the following figures represents the interval ]βˆ’1,0]?

Answer

We recall that the interval ]βˆ’1,0] represents all of the numbers between βˆ’1 and 0. It is a half-open interval and in particular includes the endpoint 0 but does not include βˆ’1. More formally, we have ]βˆ’1,0]={π‘₯βˆΆβˆ’1<π‘₯≀0,π‘₯βˆˆβ„}.

To represent this set on a number line, we want an open circle at βˆ’1 to represent that βˆ’1 is not included in the set and a closed circle at 0 to show that 0 is a member of the set. We then highlight all of the numbers between βˆ’1 and 0 with a line giving us the following.

We can see that this is option b.

Let’s now see an example of using the definitions of interval notation to determine whether a number is an element of a given interval.

Example 2: Determining Whether a Number Belongs to an Interval

Which of the following is true?

  1. 5∈√6,√26
  2. 5βˆ‰οŸβˆš6,√26

Answer

We first recall that √6,√26 is the open interval from √6 to √26; this means that it includes all of the numbers between these two values but not the endpoints themselves. In particular, we can write this as follows: √6,√26={π‘₯∢√6<π‘₯<√26,π‘₯βˆˆβ„}.

To determine whether 5 is a member of this set, we need to determine whether 5 is greater than √6 and less than √26. We can do this in a few different ways; one method is to write out the decimal expansion of the radicals. We have √6=2.449…,√26=5.099….

We can then see that 2.449…<5<5.099….

Hence, 5∈√6,√26.

Thus far, we have only dealt with limited intervals. However, we can describe unlimited subsets of the real numbers by using inequalities. For example, the positive numbers can be written as π‘₯>0. This is an unlimited (or unbounded) set, since there is no greatest element. We can represent this set on a number line as follows.

We use an open circle at 0 to show that this element is not included in the set and we can add an arrow to show that this interval extends infinitely in the positive direction.

This is not the only way we can represent this interval. We can write this in interval notation by introducing the symbol ∞, called infinity. Infinity is not a real number; however, we can think of it as being bigger than any real number. In other words, it is not finite. This allows us to represent the positive numbers in the interval notation ]0,∞[.

We can describe this concept and the idea of unlimited intervals as follows.

Definition: Unlimited Intervals

If an interval extends forever in either direction of the number line, we call it an unlimited interval. We use the symbol ∞ (read infinity) to represent something bigger than any real number. We can also use βˆ’βˆž to represent something smaller than any real number.

For any real number π‘Ž, we have the following intervals:

  • [π‘Ž,∞[={π‘₯∢π‘₯β‰₯π‘Ž,π‘₯βˆˆβ„}: this is the set of all values greater than or equal to π‘Ž.
  • ]π‘Ž,∞[={π‘₯∢π‘₯>π‘Ž,π‘₯βˆˆβ„}: this is the set of all values greater than π‘Ž.
  • ]βˆ’βˆž,π‘Ž]={π‘₯∢π‘₯β‰€π‘Ž,π‘₯βˆˆβ„}: this is the set of all values less than or equal to π‘Ž.
  • ]βˆ’βˆž,π‘Ž[={π‘₯∢π‘₯<π‘Ž,π‘₯βˆˆβ„}: this is the set of all values less than π‘Ž.

The introduction of the symbol for infinity does introduce a few key concepts that are important to note.

First, we cannot have an interval that is closed on the side of infinity. The reason for this is that βˆžβˆ‰β„, so we cannot include this in our intervals, since we are only interested in subsets of the real numbers.

Second, we can represent ℝ as an interval. The set of real numbers can be written as ]βˆ’βˆž,∞[, since this interval includes all real numbers.

Let’s now see an example of representing a given set as an unlimited interval.

Example 3: Expressing a Given Set in Interval Notation

Express the following set using interval notation 𝑋={π‘₯∢π‘₯β‰₯2,π‘₯βˆˆβ„}.

Answer

By looking at the set notation for 𝑋, we see that its elements must be real numbers and they must satisfy the inequality π‘₯β‰₯2. We can write this in interval notation by using 2 and infinity as its endpoints; we note that we want to include 2 in the interval and we should not include infinity.

This gives us [2,∞[.

In our next example, we will determine which statement is true of a given unlimited interval.

Example 4: Identifying the Relation between a Number and a Set

Which of the following is true?

  1. √49∈]5,∞[
  2. √49βˆ‰]5,∞[
  3. √49βŠ‚]5,∞[
  4. √49βŠ„]5,∞[

Answer

We first recall that the notation π΄βŠ‚π΅ means that 𝐴 is a subset of 𝐡. For this notation to be valid, both 𝐴 and 𝐡 must be sets. However, in the question, we have √49βŠ‚]5,∞[ and we know that √49 is not a set, so option C is not a correct statement.

A very similar statement is true for option D. Although √49 is not a subset of ]5,∞[, we still cannot use the notation √49βŠ„]5,∞[, since this requires √49 to be a set. Since it is not a set, we cannot compare the two objects in this way, so the answer is not D.

The remaining options raise the question of whether √49 is a member of this interval. To determine this, let’s recall what is meant by ]5,∞[. It is the set of all real numbers greater than 5; we can write this as ]5,∞[={π‘₯∢π‘₯>5,π‘₯βˆˆβ„}.

We know that √49=7, and this is greater than 5.

Hence, √49∈]5,∞[, which is option A.

Since intervals are sets, we can perform all of the set operations on intervals. For example,

  • we can take the union of intervals to represent the real numbers in either interval;
  • we can take the intersection of intervals to represent the real numbers in both intervals;
  • we can take the set difference of two intervals to remove elements of an interval from another interval;
  • we can take the complement of an interval to represent all of the real numbers not in the interval.

There are several techniques for evaluating these set operations on intervals. One method is to directly work with the inequalities. However, it can be useful to work these examples through with a number line.

For example, let’s evaluate [βˆ’1,3]βˆͺ]2,5[. We can do this by sketching both intervals on a number line. This gives us the following.

To determine their union, we want any number in either interval. We can see that this includes any number greater than or equal to βˆ’1 that is less than 5.

We can then write this as [βˆ’1,5[.

Let’s now see an example of determining an expression for a set represented graphically on a number line.

Example 5: Identifying the Interval Represented on a Given Number Line

Which of the following expressions represents the set shown on the number line?

  1. β„βˆ’]βˆ’4,1[
  2. β„βˆ’[βˆ’4,1]
  3. β„βˆ’[βˆ’4,1[
  4. β„βˆ’]βˆ’4,1]
  5. ]βˆ’βˆž,βˆ’4[βˆͺ[1,∞[

Answer

There are many different ways of representing the set given on the number line. For example, we can see that the numbers greater than βˆ’4 and less than or equal to 1 are the only ones not included in this set. We can then recall that the complement of a set means all of the elements that are not in the set. We note that the elements not in this set are the ones in the interval ]βˆ’4,1]. Hence, the complement of this set, ]βˆ’4,1], is the one represented on the number line.

However, this is not one of the given options. Instead, we note that we can think of the complement as removing elements from the entire set of real numbers. We have that ]βˆ’4,1]=β„βˆ’]βˆ’4,1], since both sets are all the real numbers that are not greater than βˆ’4 and less than or equal to 1.

We can also write this set as ]βˆ’4,1]=]βˆ’βˆž,βˆ’4]βˆͺ]1,∞[.

This highlights why option E is incorrect: the set in option E includes 1; however, 1 is not an element on the given number line.

We can see that the correct answer is option D, β„βˆ’]βˆ’4,1].

In our previous example, we showed a useful result comparing the complement of an interval and the difference with the set ℝ.

Definition: Complement of an Interval

If 𝐼 is an interval, then 𝐼=β„βˆ’πΌοŽ˜.

This is of course true for any subset of ℝ, not just intervals, but it is useful to commit this result to memory.

In our next example, we will simplify an expression of the intersection of two given intervals.

Example 6: Finding the Intersection of Two Intervals

Given that 𝑋=[βˆ’8,3[ and π‘Œ=[βˆ’4,βˆ’3[, find π‘‹βˆ©π‘Œ.

Answer

We first recall that the intersection of two sets is the set of elements in both sets. Therefore, to find π‘‹βˆ©π‘Œ, we want to find all real numbers that are in both intervals, [βˆ’8,3[ and [βˆ’4,βˆ’3[. We will find these numbers by sketching both intervals on a number line.

First, [βˆ’8,3[={π‘₯βˆΆβˆ’8≀π‘₯<3,π‘₯βˆˆβ„}: we can sketch this on a number line by drawing a closed circle at βˆ’8 to show that this endpoint is included in the set and an open circle at 3 to show that this is an element of the interval. We get the following.

We can follow the same process for [βˆ’4,βˆ’3[={π‘₯βˆΆβˆ’4≀π‘₯<βˆ’3}.

We can see that all of the interval [βˆ’4,βˆ’3[ is contained within [βˆ’8,3[. In other words, [βˆ’4,βˆ’3[βŠ‚[βˆ’8,3[. This means that their intersection is the entire subset: [βˆ’4,βˆ’3[.

Hence, [βˆ’4,βˆ’3[∩[βˆ’8,3[=[βˆ’4,βˆ’3[.

Let’s finish by recapping some of the important points from this explainer.

Key Points

  • For real numbers π‘Ž and 𝑏, where π‘Ž<𝑏, we can define the following limited intervals:
    • ]π‘Ž,𝑏[={π‘₯βˆΆπ‘Ž<π‘₯<𝑏,π‘₯βˆˆβ„}: this is called the open interval from π‘Ž to 𝑏.
    • [π‘Ž,𝑏]={π‘₯βˆΆπ‘Žβ‰€π‘₯≀𝑏,π‘₯βˆˆβ„} this is called the closed interval from π‘Ž to 𝑏.
    • ]π‘Ž,𝑏]={π‘₯βˆΆπ‘Ž<π‘₯≀𝑏,π‘₯βˆˆβ„}: this is a half-open (or half-closed) interval from π‘Ž to 𝑏.
    • [π‘Ž,𝑏[={π‘₯βˆΆπ‘Žβ‰€π‘₯<𝑏,π‘₯βˆˆβ„}: this is a half-open (or half-closed) interval from π‘Ž to 𝑏.
  • If an interval extends forever in either direction of the number line, we call it an unlimited interval. We use the symbol ∞ (read infinity) to represent something bigger than any real number. We can also use βˆ’βˆž to represent something smaller than any real number.
  • For any real number π‘Ž, we have the following unlimited intervals:
    • [π‘Ž,∞[={π‘₯∢π‘₯β‰₯π‘Ž,π‘₯βˆˆβ„}: this is the set of all values greater than or equal to π‘Ž.
    • ]π‘Ž,∞[={π‘₯∢π‘₯>π‘Ž,π‘₯βˆˆβ„}: this is the set of all values greater than π‘Ž.
    • ]βˆ’βˆž,π‘Ž]={π‘₯∢π‘₯β‰€π‘Ž,π‘₯βˆˆβ„}: this is the set of all values less than or equal to π‘Ž.
    • ]βˆ’βˆž,π‘Ž[={π‘₯∢π‘₯<π‘Ž,π‘₯βˆˆβ„}: this is the set of all values less than π‘Ž.
  • We can take the union, intersection, difference, and complement of intervals.
  • If 𝐼 is an interval, then 𝐼=β„βˆ’πΌοŽ˜.

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