1.01 Subsets of the real number system
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.
| 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.
| $\lbrace x\mid x=2n,n\in N\rbrace${x∣x=2n,n∈N} |
"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}
True
AFalse
B
Question 3
Consider the set $A=\left\{2,4,6,8\right\}$A={2,4,6,8}
Which of the following is the correct set builder notation for $A$A?
{$x$x$|$| $2\le x\le8$2≤x≤8}
A{$x$x$|$|$x$x is an even number}
B{$x$x$|$| $x$x is an even number and $x\ge2$x≥2}
C{$x$x$|$| $x$x is an even number and $2\le x\le8$2≤x≤8}
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 $\left(-\infty,\infty\right)$(−∞,∞).
The following intervals are bounded by the numbers $a$a and $b$b.
| Inequality | Interval |
|---|---|
| $a\le x\le b$a≤x≤b | $\left[a,b\right]$[a,b] |
| $a\le x |
$\left[a,b\right)$[a,b) |
| $a |
$\left(a,b\right]$(a,b] |
| $a |
$\left(a,b\right)$(a,b) |
The following intervals are considered unbounded.
| Inequality | Interval |
|---|---|
| $x\ge a$x≥a | $\left[a,\infty\right)$[a,∞) |
| $x\le a$x≤a | $\left(-\infty,a\right]$(−∞,a] |
| $x>a$x>a | $\left(a,\infty\right)$(a,∞) |
| $x |
$\left(-\infty,a\right)$(−∞,a) |
| $-\infty |
$\left(-\infty,\infty\right)$(−∞,∞) |
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.
The endpoints of this interval, from left to right, are $\editable{}$ and $\editable{}$.
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.
The endpoints of this interval are $\editable{}$ and $\editable{}$.
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.
Express the interval in set-builder notation:
{$x$x$\mid$∣$\editable{}$}
Graph the interval on the number line.