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 setbuilder 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 setbuilder
notation:
Setbuilder notation is a mathematical shorthand for
precisely stating all numbers of a specific set that
possess a specific property.

is setbuilder 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]

HalfOpen
Interval: (a, b] is interpreted
as a < x < b where a is not included,
but b is included. 
(1, 5]

HalfOpen
Interval: [a, b) is interpreted as
a
< x < b where a is included, but b is not
included. 
[1, 5)

Nonending
Interval:
is interpreted as x > a where
a is
not included and infinity is always expressed as
being "open" (not included). 

Nonending
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] 
