Subsets of Real Numbers

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intervals

We have whittled our investigation down to a study of real-valued functions. For us, these are functions whose domain and range are both subsets of the real numbers. When it comes right down to it, Precalculus is a study of our numbers. We will see in Precalculus that our numbers include more than we might think.

Naturally, we have a need to communicate about sets of real numbers. We have several ways of communicating our ideas of real numbers, both algebraically and graphically.

Building Blocks

The subsets we are interested in are collections of three types of basic building blocks.

Building Block: Individual Numbers
Our subsets might include one or more individual isolated numbers (singletons).
- Our set might be just the numbers -3, 2, and 5.
- Curly braces around a comma-separated list denotes a set of individual numbers $\{ -3, 2, 5 \}$
- Graphically, individual dots on a number line represent a set of the corresponding individual numbers.
Building Block: Finite (Bounded) Intervals
Our subsets might include whole, unbroken, pieces or segments of the real number line of a finite length. These are called finite intervals or bounded intervals. Finite intervals include ALL of the real numbers between one specific number and another specific number. Questions arise over the inclusion or exclusion of the two ends of the interval. Therefore, our communication needs to address them specifically.
- All of the real numbers between $4$ and $7$, including both $4$ and $7$. We use square brackets to indicate inclusion: $[4, 7]$
- All of the real numbers between $4$ and $7$, including $4$ but not $7$. We use parenthese to indicate exclusion: $[4, 7)$
- All of the real numbers between $4$ and $7$, not including $4$, but including $7$: $(4, 7]$
- All of the real numbers between $4$ and $7$, including neither $4$ nor $7$: $(4, 7)$
Video: Interval Notation for Bounded Intervals
[ Click on the arrow to the right to expand for the video. ]
Building Block: Infinite (Unbounded) Intervals
Our subsets might include the right or left half of the real number line. These are called infinite intervals or unbounded intervals. Infinite intervals include ALL of the real numbers less than a specific real number, or ALL of the real numbers greater than a specific real number.
- All of the real numbers less than $4$ and including $4$. When writing, we use negative infinity, -$\infty $, to indicate the interval extends without end, $(-\infty , 4]$. When drawing, we use a left arrow.
- All of the real numbers less than $4$ but not including $4$, $(-\infty , 4)$
- All of the real numbers greater than $4$ and including $4$. When writing, we use positive infinity, $\infty $, to indicate the interval extends without end, $[4, \infty )$. When drawing, we use a right arrow.
- All of the real numbers greater than $4$ but not including $4$, $(4, \infty )$.

Note: $-\infty $ and $\infty $ are not real numbers, therefore they are never included in any set of real numbers. $-\infty $ and $\infty $ are used as communication symbols. They let us know the intervals are unbounded. Therefore, they always have parentheses. Square brackets are never used for $-\infty $ or $\infty $.

Video: Interval Notation for Unbounded Intervals

[ Click on the arrow to the right to expand for the video. ]

Bounded and Unbounded

Let $S$ be a subset of the real numbers.

The set $S$ is bounded from above, if there exists a real number, $B$, such that for all $s \in S$, we have $s \leq B$. There is a real number greater than or equal to all of the numbers in $S$.
The set $S$ is unbounded from above, if for any selected real number, $B$, there exists $s \in S$, such that $B < s$. No matter how high the bar is set, $S$ contains a number greater than the bar. There is no real number that is greater than or equal to all of the numbers in $S$.
The set $S$ is bounded from below, if there exists a real number, $B$, such that for all $s \in S$, we have $B \leq s$. There is a real number less than or equal to all of the numbers in $S$.
The set $S$ is unbounded from below, if for any selected real number, $B$, there exists $s \in S$, such that $s < B$. No matter how low the bar is set, $S$ contains a number less than the bar. There is no real number that is less than or equal to all of the numbers in $S$.
The set $S$ is bounded, if it is both bounded from above and below.
The set $S$ is unbounded, if it is not bounded.

Interval Notation

In the examples above, we used interval notation to describe the intervals in writing. Interval notation adheres to a few rules.

For finite intervals, write the two extreme interval numbers in the order they would appear on the number line, separated by a comma. These are called endpoints of the intervals.
Use a square bracket to indicate an endpoint is included in the interval.
Use a parenthesis to indicate an endpoint is excluded from the interval.
For infinite (unbounded) intervals, use $-\infty $ to indicate the interval extends without end to the left. $-\infty $ always appears on the left side. $-\infty $ always has a parenthesis around it, because $-\infty $ is not a real number and thus cannot be included in an interval of real numbers.
For infinite (unbounded) intervals use $\infty $ to indicate the interval extends without end to the right. $\infty $ always appears on the right side. $\infty $ always has a parenthesis around it, because $\infty $ is not a real number and thus cannot be included in an interval of real numbers.

Which is valid interval notation?

$[3, -2)$ $[-2, 3)$

Which is valid interval notation?

$[1, \infty ]$ $[1, \infty )$

Select all that have valid interval notation.

$[-3, -5]$ $[1, 8)$ $(-\infty , -4]$ $[-\infty , 7)$ $(-3, 3)$ $(-1, 2]$ $[6, 2]$

Numbers vs. Points

Many people in many countries over many centuries have studied mathematics and invented language to communicate their findings. This is a problem for us. We are stuck with their language, even though it might not be what we would have selected. They had their reasons. We live with their reasons.

On our number line we plot dots to represent numbers and we call these points. They are points on the number line, because a point identifies a location. We will soon represent our functions with 2-dimensional drawings. We will still have points and they will still identify a location. However, with two dimensions, describing the location will require two numbers, rather than one. A point will no longer represent a single number.

Our language will become a little blurred when we talk about points versus numbers. Just be aware that our language is old and has been twisted to accommodate many thoughts by many people.

We will attempt to keep the blur to a miminum in this course by our judicial use of the words number and point.

Note: We will reserve the word “point” for graphical descriptions and “number” for algebraic communication.

Membership

Now that we can describe sets of numbers with interval notation, we can use our membership symbol, $\in $, to communicate when individual numbers are included in the interval or when they are not included, $\notin $.

Which is a valid statement?

$ 2 \in [3, -2)$ $-2 \in [-2, 3)$

Select all valid statements.

$4 \in [-3, -5]$ $4 \in [1, 8)$ $5 \notin (-\infty , -4]$ $0 \notin [-\infty , 7)$ $0 \in (-3, 3)$ $0 \in (-1, 2]$ $0 \in [6, 2]$ $3 \notin [1, \infty ]$ $-3 \notin [1, \infty )$

Open Intervals

Intervals of the form $(a, b)$, $(-\infty , b)$, or $(a, \infty )$ are called open intervals.

Closed Intervals

Intervals of the form $[a, b]$, $(-\infty , b]$, or $[a, \infty )$ are called closed intervals.

$(-\infty , \infty )$ is both an open interval and a closed interval.

The Empty Set

The empty set is the set containing no numbers. $\emptyset $ and $\{ \, \}$ are both symbols for the empty set.

The empty set, $\emptyset $, is both an open interval and a closed interval.

Set Builder Notation

A second way to represent intervals in writing is with inequalities. However, we cannot simply insert an inequality into our description, because people use inequalities in several contexts. So, we need some notation that tells people we are describing a set. Set builder notation includes shorthand symbols to represent sentences such as

“The set of all real numbers such that the numbers are greater than or equal to 4 and less than 7.”

We begin with a pair of curly braces $\{ \, \}$ containing a vertical bar: $|$. The vertical bar separates two areas inside the curly braces. The left area is where we describe the types of numbers we are working with (real, integers, rational, etc). The right area is where we place our inequality.

Video: Interval Notation, Inequalities, and Graphs

[ Click on the arrow to the right to expand for the video. ]

Set-Builder Notation

“The set of all real numbers such that the numbers are greater than or equal to 4 and less than 7.”

\[ \{ r \in \mathbb {R} \, | \, (4 \leq r) \text { and } (r < 7) \} \]

\[ \{ r \in \mathbb {R} \, | \, 4 \leq r < 7 \} \]

ALL real numbers would include numbers such as $5$, $\frac {17}{4}$, $\sqrt {19}$, $6.354987263$, and $211^{\tfrac {1}{\pi }}$, not just whole numbers.

Set-Builder Notation

“The set of all integers such that the numbers are greater than or equal to 4 and less than 7.”

\[ \{ n \in \mathbb {Z} \, | \, 4 \leq n < 7 \} \]

explanation

This would be the set $\{ 4, 5, 6 \}$, since $n \in \mathbb {Z}$.

Select the interval notation that describes the same set as

\[ \{ t \in \mathbb {R} \, | \, 2 < t \leq 9 \} \]

$[9, 2)$ $[2, 9)$ $(2, 9]$ $[2, 9]$ $(2, 9)$

Complete the interval notation that describes the same set as

\[ \{ r \in \mathbb {R} \, | \, -3 \leq r < 7 \} \]

\[ \left [ \answer {-3} , \answer {7} \right ) \]

Gluing Building Blocks Together: Unions

Our building blocks for sets of real numbers are lists of individual numbers along with intervals. We can combine these to describe any set of real numbers that we need. These combinations are called unions. The union symbol is $\cup $. Unions are usually written with interval notation.

Union

The union of two sets is a set containing all elements from both sets.

\[ A \cup B = \{ x \, | \, (x \in A) \, \text { or } \, (x \in B) \} \]

Intersection

The intersecton of two sets is a set containing all elements common in both sets.

\[ A \cap B = \{ x \, | \, (x \in A) \, \text { and } \, (x \in B) \} \]

Video: Sets

[ Click on the arrow to the right to expand for the video. ]

Video: Singletons

[ Click on the arrow to the right to expand for the video. ]

Interval Notation

\[ [-4, -2] \cup \{ 2 \} \cup (3, 6] \]

Interval Notation

\[ (-\infty , -1] \cup (4, 8) \]

Interval Notation

\[ (-7, -4] \cup \{ -1, 1 \} \cup (3, \infty ) \]

Select the interval notation that describes the same set as the following graph.

$[-3, 2) (4, 7)$ $[-3, 2) \cup (4, 7)$

Reduced Form

All of the following describe the same set.

$[-2, 2)$
$[-2, -1] \cup (-1, 2)$
$[-2, -1) \cup [-2, 0) \cup [-2, 1) \cup [-2, 2)$
$[-2, 0) \cup \{ 0 \} \cup (0, 2)$
$[-2, 0) \cup \{ 0 \} \cup (0, 1) \cup \{ 1 \} \cup (1, 2)$
$[-2, 0) \cup [-1, 1) \cup [1, 2)$

Having such a wide variety of descriptions is very valuable as we move through mathematics. However, for communication purposes, it causes confusion. Therefore, we are establishing an official reduced form for interval notation. We will always use the reduced form, unless there is a good reason not to. (There are often good reasons.)

Our reduced form will adhere to some rules:

Reduced Interval Notation

Do not use overlapping or intersecting intervals.
Do not list individual numbers that are members of an interval.
Do not list individual numbers that can be used as endpoints of an interval.
Write the intervals and sets of individual numbers in proper numeric order.

Communication!

Our reduced form of interval notation mimics exactly what you would draw on a number line.

Video: Reduced Interval Notation

[ Click on the arrow to the right to expand for the video. ]

Select all of the unions that are written in reduced form.

$(-3, 2) \cup (4, 7)$ $[-3, 4) \cup (2, 7)$ $(-3, 2] \cup [4, 7] \cup \{ 3 \}$ $[-3, 2) \cup \{ 3 \} \cup (4, 7)$ $(4, 7) \cup [-3, 2)$

Maximal Intervals

When a set of real numbers is written in reduced form, then each of the intervals is called a maximal interval of the set.

Maximal intervals cannot be made larger inside the set.

Maximal Intervals

Let $S = [-3, 2) \cup \{ 3 \} \cup (4, 7)$, then

$[-3, 2)$ is a maximal interval of $S$.
$(4, 7)$ is a maximal interval of $S$.

$3$ is an isolated number of $S$.

$-3$, $2$, $4$, and $7$ are all endpoints of maximal intervals of $S$.

$-3$ is the only included endpoint of a maximal interval of $S$.

Endpoints are usually the focus of maximal intervals.

nonMaximal Intervals

Let $S = [-4, 1) \cup \{ 1, 4 \} \cup (5, 9)$, then

$(-3, 1)$ is not a maximal interval of $S$.

We can extend this interval to $[-3, 1)$ and still stay inside $S$.

Of course, we could have also extended $(-3, 1)$ to the maximal interval it sits inside, $(-3, 1) \subset [-4, 1)$.

Maximal intervals give us information about the structure of the domain of functions.

Our official reduced interval notation uses maximal intervals.

Dots

We use square brackets and parentheses for both written algebaic notation and graphical notation.

We have a very popular alternative for the graphical notation.

solid or closed dots are often used in place of square brackets to show a number is included in an interval.
hollow or open dots are often used in place of parentheses to show a number is not included in an interval.

Closed Interval