|
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.
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. |
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.
|
|
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 x 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
 |
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) |
(diagram) |
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 a < 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
a
< x < b where a is included, but b is not
included. |
[1, 5)
 |
Non-ending
Interval:
is interpreted as x > a where
a is
not included and infinity is always expressed as
being "open" (not included). |
 |
Non-ending
Interval:
is interpreted as x < b
where
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 "U",
which 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: |
|
WORDS: |
SYMBOLS: |
|
"all
numbers between positive one and positive five,
including the one and the five." |
1 <
x <
5 |
|
"x is less than or equal to 5 and greater than or equal
to 1" |
{ x
|
1 <
x <
5} |
|
"x is between 1 and 5, inclusive" |
[1,5] |
|
= real numbers;
= integer numbers;
= natural numbers.