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1.01 Subsets of the real number system

Lesson

Recall that a set is a list of objects, each of which is called an element. Let's review two types of notations we can use to refer to a subset of the real numbers.

Set builder notation

Set builder notation follows a common convention.

Set builder notation
Symbols $\lbrace x\mid${x $x>5\rbrace$x>5}
Read The set of all numbers $x$x such that $x$x has a certain property

Many subsets of the real numbers have abbreviations which are commonly used.

Set Letter Abbreviation
Real numbers $R$R
Rational numbers $Q$Q
Irrational numbers $I$I
Integers $Z$Z
Whole numbers $W$W
Natural numbers $N$N
Empty set $\varnothing$

Worked example

Question 1

Describe the set of even numbers using set builder notation.

Think: The set of even numbers are all natural numbers that are multiples of $2$2. If $x$x is an even number, then $x=2n$x=2n, where $n$n is a natural number.

Do: Translate to set builder notation.

$\lbrace x\mid x=2n,n\in N\rbrace${xx=2n,nN}

 "the set of all x such that $x=2n$x=2n, where $n$n is an element of the natural numbers"

 

Practice questions

Question 2

Is the following statement true or false?

$1$1$\in$$\left\{9,8,6,5,1\right\}${9,8,6,5,1}

  1. True

    A

    False

    B

Question 3

Consider the set $A=\left\{2,4,6,8\right\}$A={2,4,6,8}

  1. Which of the following is the correct set builder notation for $A$A?

    {$x$x$|$| $2\le x\le8$2x8}

    A

    {$x$x$|$|$x$x is an even number}

    B

    {$x$x$|$| $x$x is an even number and $x\ge2$x2}

    C

    {$x$x$|$| $x$x is an even number and $2\le x\le8$2x8}

    D

 

Interval notation

An interval on the real number line is the set of numbers between two endpoints. One or both endpoints, or neither, can belong to interval and there are notations for each possibility.

Intervals that include their endpoints are called closed intervals. To specify a closed interval including all numbers between $-1$1 and $2\pi$2π, for example, we write $\left[-1,2\pi\right]$[1,2π]. The symbols $[$[ and $]$] indicate that the numbers $-1$1 and $2\pi$2π are considered to belong to the interval.

Intervals that do not include their endpoints are called open intervals. The notation for these uses parenthesis. For example, we write $\left(0,100\right)$(0,100) to mean the set of numbers between $0$0 and $100$100 but not including either $0$0 or $100$100

It is possible for an interval to be closed at one end but open at the other. For example, $[0,\sqrt{2})$[0,2) or $(-9,0]$(9,0].

An interval with no upper bound is indicated with the $\left[a,\infty\right)$[a,) sign. Such intervals are said to be open on the right. Similarly, an interval with no lower bound is open on the left and is notated with the sign $\left(-\infty,b\right]$(,b]

Thus, for example, we can indicate the whole real number line with the notation $\left(-\infty,\infty\right)$(,).

Interval notation

The following intervals are bounded by the numbers $a$a and $b$b.

Inequality Interval
$a\le x\le b$axb $\left[a,b\right]$[a,b]
$a\le xax<b $\left[a,b\right)$[a,b)
$aa<xb $\left(a,b\right]$(a,b]
$aa<x<b $\left(a,b\right)$(a,b)

The following intervals are considered unbounded.

Inequality Interval
$x\ge a$xa $\left[a,\infty\right)$[a,)
$x\le a$xa $\left(-\infty,a\right]$(,a]
$x>a$x>a $\left(a,\infty\right)$(a,)
$xx<a $\left(-\infty,a\right)$(,a)
$-\infty<x< $\left(-\infty,\infty\right)$(,)

Previously, we looked a solving inequalities. We can be asked to give our answers in either inequality or interval notation. Be sure to read the question carefully.

 

Practice questions

Question 4

Consider the pictured inequality.

A horizontal number line is shown, with integers labeled from -3 to 7. On this line, a close dot is located directly above the number 2, and an open dot is above the number -1. A solid line connects the solid point above the number 2 to the hollow point above the number -1.
  1. The endpoints of this interval, from left to right, are $\editable{}$ and $\editable{}$.

  2. Which of the following is the correct notation for the pictured interval?

    $[$[$-1$1,$2$2$]$]

    A

    $($($-1$1,$2$2$]$]

    B

    $[$[$-1$1,$2$2$)$)

    C

    $($($-1$1,$2$2$)$)

    D

Question 5

Consider the inequality $-7$7$\le$$x$x$\le$$3$3.

  1. The endpoints of this interval are $\editable{}$ and $\editable{}$.

  2. Which of the following is the correct notation for the given inequaliaty?

    $[$[$-7$7,$3$3$]$]

    A

    $($($-7$7,$3$3$]$]

    B

    $($($-7$7,$3$3$)$)

    C

    $[$[$-7$7,$3$3$)$)

    D

Question 6

Consider the interval $\left(8,9\right]$(8,9] and answer the following questions.

  1. Express the interval in set-builder notation:

    {$x$x$\mid$$\editable{}$}

  2. Graph the interval on the number line.

    0510

 

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