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Section1.7Notation for Intervals

If you say

\begin{equation*} (\text{age of a voter})\geq18 \end{equation*}

and have a particular voter in mind, what is that person's age? There's no way to know for sure. Maybe they are \(18\text{,}\) but maybe they are older. It's helpful to visualize the possibilities with a number line, as in Figure 1.7.1.

a number line, where the number a8 is marked; the portion of the number line to the right of 18 has a thick line overlaying it with an arrow pointing to the right; at 18, there is a left bracket character; text indicates the right area represents possibilities for age
Figure1.7.1\((\text{age of a voter})\geq18\)

The shaded portion of the number line in Figure 1.7.1 is a mathematical interval. For now, that just means a collection of certain numbers. In this case, it's all the numbers \(18\) and above.

It's one thing to say \((\text{age of a voter})\geq18\text{,}\) and another thing to discuss all the shaded numbers in the interval in Figure 1.7.1. In mathematics,

\begin{equation*} (\text{age of a voter})\geq18 \end{equation*}

is saying that there is one age under consideration and all we know is that it's 18 or larger. It's subtle, but this is not the same thing as the collection of all numbers that are \(18\) or larger. Mathematics has two common ways to write down these kinds of collections.

Definition1.7.2Set-Builder Notation

Set-builder notation attempts to directly say the condition that numbers in the interval satisfy. In general, write set-builder notation like:

\begin{equation*} \left\{x\mid\text{condition on }x\right\} \end{equation*}

and read it out loud as “the set of all \(x\) such that ….” For example,

\begin{equation*} \left\{x\mid x\geq18\right\} \end{equation*}

is read out loud as “the set of all \(x\) such that \(x\) is greater than or equal to \(18\text{.}\)” The breakdown is as follows.

\(\highlight{\{}\lowlight{x\mid x\geq18}\highlight{\}}\) the set of
\(\lowlight{\{}\highlight{x}\lowlight{{}\mid x\geq18\}}\) all \(x\)
\(\lowlight{\{x}\highlight{{}\mid{}}\lowlight{x\geq18\}}\) such that
\(\lowlight{\{x\mid{}}\highlight{x\geq18}\lowlight{\}}\) \(x\) is greater than or equal to \(18\)
Definition1.7.3Interval Notation

Interval notation tries to just say the numbers where the interval starts and stops. For example, in Figure 1.7.1, the interval starts at \(18\text{.}\) To the right, the interval extends forever and has no end, so we use the \(\infty\) symbol (meaning "infinity"). This particular interval is denoted:

\begin{equation*} [18,\infty) \end{equation*}

Why use “\([\)” on one side and “\()\)” on the other? The square bracket tells us that \(18\) is part of the interval and the round parenthesis tells us that \(\infty\) is not part of the interval. 1 And how could it be, since \(\infty\) is not even a number?

In general there are four types of infinite intervals. Take note of the different uses of round parentheses and square brackets.

a number line with a mark at a; the portion of the number line from a to the right has a thick line overlaying it; there is a left parenthesis at a and an arrow on the right pointing right
a number line with a mark at a; the portion of the number line from a to the right has a thick line overlaying it; there is a left bracket at a and an arrow on the right pointing right
Figure1.7.4An open, infinite interval denoted by \((a,\infty)\) means all numbers \(a\) or larger, not including \(a\text{.}\)
Figure1.7.5A closed, infinite interval denoted by \([a,\infty)\) means all numbers \(a\) or larger, including \(a\text{.}\)
a number line with a mark at a; the portion of the number line from a to the left has a thick line overlaying it; there is a right parenthesis at a and an arrow on the left pointing left
a number line with a mark at a; the portion of the number line from a to the left has a thick line overlaying it; there is a right bracket at a and an arrow on the left pointing left
Figure1.7.6An open, infinite interval denoted by \((-\infty,a)\) means all numbers \(a\) or smaller, not including \(a\text{.}\)
Figure1.7.7A closed, infinite interval denoted by \((-\infty,a]\) means all numbers \(a\) or smaller, including \(a\text{.}\)
Exercise1.7.8Interval and Set-Builder Notation from Number Lines

Subsection1.7.1Exercises

1

For each interval expressed in the number lines, give the interval notation and set-builder notation.

  1. In set-builder notation:

    In interval notation:

  2. In set-builder notation:

    In interval notation:

  3. In set-builder notation:

    In interval notation:

2

For each interval expressed in the number lines, give the interval notation and set-builder notation.

  1. In set-builder notation:

    In interval notation:

  2. In set-builder notation:

    In interval notation:

  3. In set-builder notation:

    In interval notation:

3

Here is a graph of an interval.

Write this inequality in set-builder notation:

Write this inequality in interval notation:

4

Here is a graph of an interval.

Write this inequality in set-builder notation:

Write this inequality in interval notation:

5

Here is a graph of an interval.

Write this inequality in set-builder notation:

Write this inequality in interval notation:

6

Here is a graph of an interval.

Write this inequality in set-builder notation:

Write this inequality in interval notation:

7

Here is a graph of an interval.

Write this inequality in set-builder notation:

Write this inequality in interval notation:

8

Here is a graph of an interval.

Write this inequality in set-builder notation:

Write this inequality in interval notation:

9

Here is a graph of an interval.

Write this inequality in set-builder notation:

Write this inequality in interval notation:

10

Here is a graph of an interval.

Write this inequality in set-builder notation:

Write this inequality in interval notation:

Convert set-builder notation to interval notation.

11

Change the following inequality from set-builder notation to interval notation:

\(\displaystyle{ \{ x \mid {{x}} \leq -9 \} }\)

The interval notation is .

12

Change the following inequality from set-builder notation to interval notation:

\(\displaystyle{ \{ x \mid {{x}} \leq -6 \} }\)

The interval notation is .

13

Change the following inequality from set-builder notation to interval notation:

\(\displaystyle{ \{ x \mid {{x}} \geq -4 \} }\)

The interval notation is .

14

Change the following inequality from set-builder notation to interval notation:

\(\displaystyle{ \{ x \mid {{x}} \geq -2 \} }\)

The interval notation is .

15

Change the following inequality from set-builder notation to interval notation:

\(\displaystyle{ \{ x \mid{{x}} \lt 1 \} }\)

The interval notation is .

16

Change the following inequality from set-builder notation to interval notation:

\(\displaystyle{ \{ x \mid{{x}} \lt 3 \} }\)

The interval notation is .

17

Change the following inequality from set-builder notation to interval notation:

\(\displaystyle{ \{ x \mid {{x}} \gt 5 \} }\)

The interval notation is .

18

Change the following inequality from set-builder notation to interval notation:

\(\displaystyle{ \{ x \mid {{x}} \gt 8 \} }\)

The interval notation is .

19

Change the following inequality from set-builder notation to interval notation:

\(\displaystyle{ \{ x \mid {10} \gt {x} \} }\)

The interval notation is .

20

Change the following inequality from set-builder notation to interval notation:

\(\displaystyle{ \{ x \mid {-9} \gt {x} \} }\)

The interval notation is .

21

Change the following inequality from set-builder notation to interval notation:

\(\displaystyle{ \{ x \mid {-6} \geq {x} \} }\)

The interval notation is .

22

Change the following inequality from set-builder notation to interval notation:

\(\displaystyle{ \{ x \mid {-4} \geq {x} \} }\)

The interval notation is .

23

Change the following inequality from set-builder notation to interval notation:

\(\displaystyle{ \{ x\mid{-2} \leq {x} \} }\)

The interval notation is .

24

Change the following inequality from set-builder notation to interval notation:

\(\displaystyle{ \{ x\mid{1} \leq {x} \} }\)

The interval notation is .

25

Change the following inequality from set-builder notation to interval notation:

\(\displaystyle{ \{ x \mid {3} \lt {x} \} }\)

The interval notation is .

26

Change the following inequality from set-builder notation to interval notation:

\(\displaystyle{ \{ x \mid {5} \lt {x} \} }\)

The interval notation is .

27

Change the following inequality from set-builder notation to interval notation:

\(\displaystyle{ \left\{ x \mid {{{\frac{9}{10}}}} \lt {x} \right\} }\)

The interval notation is

28

Change the following inequality from set-builder notation to interval notation:

\(\displaystyle{ \left\{ x \mid {{{\frac{10}{7}}}} \lt {x} \right\} }\)

The interval notation is

29

Change the following inequality from set-builder notation to interval notation:

\(\displaystyle{ \left\{ x \mid {{x}} \leq {-{\frac{1}{4}}} \right\} }\)

The interval notation is

30

Change the following inequality from set-builder notation to interval notation:

\(\displaystyle{ \left\{ x \mid {{x}} \leq {-{\frac{2}{7}}} \right\} }\)

The interval notation is

31

Change the following inequality from set-builder notation to interval notation:

\(\displaystyle{ \{ x \mid {x} \leq 0 \} }\)

The interval notation is .

32

Change the following inequality from set-builder notation to interval notation:

\(\displaystyle{ \{ x \mid 0 \lt {x} \} }\)

The interval notation is .