intervals
[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
- Graphically, individual dots on a number line represent a set of the
corresponding individual numbers.
- Our set might be just the numbers -3, 2, and 5.
- 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 and , including both and . We use square
brackets to indicate inclusion:
- All of the real numbers between and , including but not . We use
parenthese to indicate exclusion:
- All of the real numbers between and , including but not :
- All of the real numbers between and , including neither nor :
- All of the real numbers between and , including both and . We use square
brackets to indicate inclusion:
- 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 and including . When writing, we use
negative infinity, -, to indicate the interval extends without end, . When
drawing, we use a left arrow.
- All of the real numbers less than but not including ,
- All of the real numbers greater than and including . When writing, we use
positive infinity, , to indicate the interval extends without end, . When
drawing, we use a right arrow.
- All of the real numbers greater than but not including , .
- All of the real numbers less than and including . When writing, we use
negative infinity, -, to indicate the interval extends without end, . When
drawing, we use a left arrow.
Note: and are not real numbers, therefore they are never included in any set of real numbers. and are used as communication symbols. They let us know the intervals are unbounded.
[ Click on the arrow to the right to expand for the video. ]
Let be a subset of the real numbers.
- The set is bounded from above, if there exists a real number, , such
that for all , we have . There is a real number greater than or equal to all
of the numbers in .
- The set is unbounded from above, if for any selected real number, ,
there exists , such that . No matter how high the bar is set, contains a
number greater than the bar.
- The set is bounded from below, if there exists a real number, , such
that for all , we have . There is a real number less than or equal to all of
the numbers in .
- The set is unbounded from below, if for any selected real number, ,
there exists , such that . No matter how low the bar is set, contains a
number less than the bar.
- The set is bounded, if it is both bounded from above and below.
- The set 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 intervals, use to indicate the interval extends without end to the left. always appears on the left side. always has a parenthesis around it, because is not a real number and thus cannot be included in an interval of real numbers.
- Use to indicate the interval extends without end to the right. always appears on the right side. always has a parenthesis around it, because is not a real number and thus cannot be included in an interval of real numbers.
Many people in many countries over many centuries have studied mathematics and invented language to communicate their findings. This is a problem for us.
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.
Now that we can describe sets of numbers with interval notation, we can use our membership symbol, , to communicate when individual numbers are included in the interval or when they are not included, .
The empty set is the set containing no numbers. and are both symbols for the empty set.
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.
[ Click on the arrow to the right to expand for the video. ]
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 . Unions are usually written with interval notation.
Reduced Form
All of the following describe the same set.
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:
- 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.
Our reduced form of interval notation mimics exactly what you would draw on a number line.
When a set of real numbers is written in reduced form, then each of the intervals is called a maximal interval of the set.
Let , then
- is a maximal interval of .
- is a maximal interval of .
is an isolated number of ,
, , , and are all endpoints of maximal intervals of .
is the only included endpoint of a maximal interval of .
Let , then
is not a maximal interval of .
We can extend this interval to and still stay inside .
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