Set-builder & Interval Notation

What is Set-builder & Interval Notation?

A set is a collection of unique elements. Elements in a set do not “repeat”. For more information on sets, see Working with Sets.

Methods of Describing Sets:

Sets may be described in many ways: by roster, by set-builder notation, by interval notation, by graphing on a number line, and/or by Venn diagrams. For graphing on a number line, see Linear Inequalities. For Venn diagrams, see Working with Sets and Venn Diagrams.

What is a Roster?

By roster: A roster is a list of the elements in a set, separated by commas and surrounded by French curly braces.

is a roster for the set of integers from 2 to 6, inclusive.

is a roster for the set of positive integers.  The three dots indicate that the numbers continue in the same pattern indefinitely.
(Those three dots are called an ellipsis.)

Rosters may be awkward to write for certain sets that contain an infinite number of entries.

What is Set-Builder Notation?

By set-builder notation: Set-builder notation is a mathematical shorthand for precisely stating all numbers of a specific set that possess a specific property.

 = real numbers;   = integer numbers;   = natural numbers.


is set-builder notation for the set of integers from 2 to 6, inclusive.        = “is an element of”
Z = the set of integers
| = the words “such that”
The statement is read, “all x that are elements of the set of integers, such that, x is between 2 and 6 inclusive.”
The statement is read, “all that are elements of the set of integers, such that, the x values are greater than 0, positive.”
(The positive integers can also be indicated as the set  Z+.)
   It is also possible to use a colon ( : ), instead of the | , to represent the words “such that”.
   is the same as  

What is Interval Notation?

By interval notation: An interval is a connected subset of numbers. Interval notation is an alternative to expressing your answer as an inequality. Unless specified otherwise, we will be working with real numbers.

When using interval notation, the symbol:
( means “not included” or “open”.
[ means “included” or “closed”.
as an inequality.
in interval notation.

The chart below will show you all of the possible ways of utilizing interval notation.

Interval Notation:  (description)


Open Interval:   (a, b)  is interpreted as a < x < b  where the endpoints are NOT included.
(While this notation resembles an ordered pair, in this context it refers to the interval upon which you are working.)

(1, 5)

Closed Interval:  [a, b]  is interpreted as < x < b  where the endpoints are included.

[1, 5]

Half-Open Interval:  (a, b]  is interpreted as a < x < b where a is not included, but b is included.

(1, 5]

Half-Open Interval:  [a, b) is interpreted as < x < bwhere a is included, but b is not included.

[1, 5)

Non-ending Interval:   is interpreted as x > a wherea is not included and infinity is always expressed as being “open” (not included).

Non-ending Interval:   is interpreted as < bwhere b is included and again, infinity is always expressed as being “open” (not included).

For some intervals it is necessary to use combinations of interval notations to achieve the desired set of numbers. Consider how you would express the interval “all numbers except 13”.

As an inequality:       x < 13  or   x > 13
In interval notation: 

Notice that the word “or” has been replaced with the symbol “Uwhich stands for “union”.

Consider expressing in interval notation, the set of numbers which contains all numbers less than 0 and also all numbers greater than 2 but less than or equal to 10.

As an inequality:       x < 0   or   2 < x < 10
In interval notation: 

As you have seen, there are many ways of representing the same interval of values. These ways may include word descriptions or mathematical symbols.

The following statements and symbols ALL represent the same interval:

“all numbers between positive one and positive five, including the one and the five.” < x < 5
x is less than or equal to 5 and greater than or equal to 1″ x    | < x < 5}
x is between 1 and 5, inclusive” [1,5]