intervals

We have whittled our investigation down to a study of real-valued functions. These are functions whose domain and range are both subsets of the real numbers. When it comes right down to it, Precalculus 1 is a study of the real numbers.

[Precalculus 2 will be a study of the Complex Numbers.]

As a consequence, 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.

    • 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 \(7\), including \(7\) but not \(4\): \((4, 7]\)

    • All of the real numbers between \(4\) and \(7\), including neither \(4\) nor \(7\): \((4, 7)\)

  • 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 \(\infty \).

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.
\([-3, -5]\) \([1, 8)\) \((-\infty , -4]\) \([-\infty , 7)\) \((-3, 3)\) \((-1, 2]\) \([6, 2]\)
Which is a valid statement?
\( 2 \in [3, -2)\) \(-2 \in [-2, 3)\)
Which is a valid statement?
\(3 \notin [1, \infty ]\) \(-3 \notin [1, \infty )\)
Select all valid statements.
\(4 \in [-3, -5]\) \(4 \in [1, 8)\) \(5 \notin (-\infty , -4]\) \(0 \notin [-\infty , 7)\) \(0 \in (-3, 3)\) \(0 \in (-1, 2]\) \(0 \in [6, 2]\)

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.

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.

Select the interval notation that describes the same set as the following graph.
\((-3, 2) \cup (4, 7)\) \([-3, 2) (4, 7)\) \((-3, 2] \cup [4, 7]\) \([-3, 2) \cup (4, 7)\) \([-3, 2), (4, 7)\)

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:

Communication!

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

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 cannot be made larger inside the set.

Endpoints are usually the focus of maximal intervals.

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.

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more examples can be found by following this link
More Examples of Real-Valued Functions

2025-01-09 20:24:12