Download SETS

WRITING SETS
Ways to Write Sets:
Roster or a list of the actual elements (within brackets)
Rule: set described in words (within brackets)
Set Builder Notation similar to rule but more formal
Interval Notation used if a set is continuous and could be also
represented as an inequality
Roster Form
{2, 3, 5, 7}
{i, m, p, s}
Note: The letters are not repeated.
{California, Oregon, Washington, Alaska, Hawaii}
Graphing a line (inequality) is a visual way to represent a
continuous number set
Venn diagram used with multiple set where parts of the sets
overlap
If there is an extensive
number of elements in a set,
the rule form is more practical
than the roster form.
Writing a very large roster list
would be too time-consuming.
Rule Form
{one-digit prime numbers}
{the letters of the word Mississippi}
{states of the U.S. that touch the Pacific Ocean}
Set Builder Notation: A way of describing a set in
Interval Notation: An interval is a connected subset of
“mathematical shorthand” without listing the elements of the set. Set
builder notation is similar to rule form, but it is considered to be a
more precise and “formal” way to describe a set.
numbers. Inequalities are examples of interval subsets.
Interval notation is an alternate way to write an inequality
instead of using the symbols , , , or  , or graphing it on the
number line.
Example (1): Describing the set of all the natural (counting numbers).
Set Builder Notation: {x x  Ν } or {x : x  Ν }
Translation: “all x , such that, x is an element of the natural
numbers”
Example (2): Describing the set of multiples of 5.
Set Builder Notation: {m m is a multiple of 5} or
{m : m is a multiple of 5}
Translation: “all m, such that, m is a multiple of 5”
Note: Use the “not included” symbol
when dealing with infinity and negative
infinity since you can’t ever reach the
end of either.
Symbols used in interval notation:
( or ) means “not included in the set”
[ or ] means “included in the set”
- ∞ means “negative infinity”
∞ means “positive infinity”
Example (1): The inequality 2  x  6
Interval Notation: [ 2, 6 )
Translation: “all real numbers in the interval of 2 to 6, including
2 and excluding 6”
Example (2): The inequality x  5
Interval Notation: ( 5, ∞)
Translation: “all real numbers greater than 5”
Interval Notation: (description)
(graphic)
Open Interval: (1, 5)
 is the inequality 1 < x < 5
 where the endpoints are NOT included.
(1, 5)
Closed Interval: [1, 5]
 is the inequality 1 < x <5
 where the endpoints are included.
[1, 5]
Half-Open Interval: (1, 5]
 is the inequality 1 < x < 5
 where 1 is not included, but 5 is included.
(1, 5]
Half-Open Interval: [1, 5)
 is the inequality 1 < x < 5
 where 1 is included, but 5 is not included.
[1, 5)
Non-ending Interval: (1,)
 is the inequality x > 1
 where 1 is not included
 infinity is always expressed as being "open" (not
included).
Non-ending Interval: (-, 5]
 is the inequality x < 5
 where 5 is included
 infinity is always expressed as being "open" (not
included).
You may see a set written in any of the formats we have discussed.
The following is an example of the same exact set written:




In words
As an inequality
In set builder notation
In interval notation
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."
Inequality: 1 < x < 5
"x is less than or equal to 5 and greater than or
equal to 1"
Set Builder: { x
"x is between 1 and 5, inclusive"
Interval: [1,5]
| 1 < x < 5}