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Foundations for Functions
A-1 Sets of Real Numbers
Objectives
Classify and graph real numbers
Represent sets of numbers using interval
notation and set-builder notation.
Essential Question
How can you tell if a number belongs
to the set of rational numbers?
A set is a collection of items called elements. The
rules of 8-ball divide the set of billiard balls into
three subsets: solids (1 through 7), stripes (9
through 15) and the 8 ball. A subset is a set
whose elements all belong to another set. The
empty set, denoted Ø, is a set containing no
elements.
All of the numbers that you use in
everyday life are real numbers. Each real
number corresponds to exactly one point
on the real number line, and every point
on the number line represents exactly one
real number. Numbers increase from left
to right on the real number line.
Real numbers can be classified as either
rational or irrational.
Rational Numbers
A rational number can be expressed as a
quotient ( or ratio) of two integers where
the denominator is not zero. The decimal
form of a rational number is either a
terminating or repeating decimal.
1
2
19
Examples: = 0.5 ,
10
3
2
−3 = − = −3.0, 4 =
1
1
=1.9,
4
−
3
= −1.333…,
0
1
= 2.0, 0 = = 0
Irrational Numbers
A real number that is not rational is irrational.
An irrational number cannot be expressed
as a quotient of two integers, and their
decimal forms do not terminate or repeat.
However, you can approximate these
numbers using terminating decimals to a
specific place value using the symbol ≈.
Examples:
5, 𝜋, 0.1010010001…
The sets of natural numbers, {1, 2, 3, 4, 5, …},
whole numbers, {0, 1, 2, 3, 4, …}, and
integers, {…, -3, -2, -1, 0, 1, 2, 3, …} are all
subsets of the rational numbers. Each one
is a subset of the rational numbers because
every number in the above sets can be
written as a quotient of integers with a
denominator of 1.
REAL (R)
IRRATIONAL (I)
𝜋, − 2, 0.2465123…
RATIONAL (Q)
¼, -0.25, 0.333…, 1½
INTEGERS (Z)
…, -5, -4, -3, -2, -1
WHOLE (W)
0
NATURAL (N)
1, 2, 3, 4, …
Example 1
Name the sets of numbers to which
and graph on a number line.
Answer: rationals (Q) and reals (R)
−
4
3
•
−3
−2
−1
0
1
2
3
4
−
3
belongs
Your Turn
Name the sets of numbers to which 6
belongs. Graph on a number line.
Answer: irrationals (I) and reals (R)
Notice that 6 is between 2 and 3, but closer to 2 since 2² = 4
and 3² = 9 and 6 is closer to 4 than 9.
6
•
−3
−2
−1
0
1
2
3
There are many ways to represent sets. For
instance, you can use words to describe a set.
You can also use roster (set) notation, in which
the elements of a set are listed between
braces, { }.
Roster (Set)
Words
Notation
The set of billiard
balls is numbered 1
through 15.
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15}
A set can be finite like the set of billiard ball
numbers or infinite like the natural numbers
{1, 2, 3, 4, …}. A finite set has a definite, or
finite, number of elements. An infinite set has
an unlimited, or infinite, number of elements.
Many infinite sets, such as the real numbers,
cannot be represented in roster notation.
There are other methods of representing
these sets.
As previously stated, the number line represents
the set of all real numbers.
For example, the set of real numbers between
3 and 5, which is also an infinite set, can be
represented on a number line or by an
inequality, 3 < x < 5
ο
−1
0
1
2
3
ο
4
5
An interval is the set of all numbers between
two endpoints, such as 3 and 5. In interval
notation the symbols [ and ] are used to
include an endpoint in an interval, and the
symbols ( and ) are used to exclude an
endpoint from an interval.
( 3, 5 )
The set of real numbers between but
not including 3 and 5
An interval that extends forever in the positive
direction goes to infinity (∞), and an interval
that extends forever in the negative direction
goes to negative infinity (−∞). Because ∞ and
−∞ are not numbers, they cannot be included
in a set of numbers, so parentheses are used
to enclose them in an interval.
∞
−∞
−3
−2
−1
0
1
2
3
Methods of Representing Intervals
Words
Numbers less than
3
Numbers greater
than or equal to
−2
Numbers
between 2 and 4
Numbers 1
through 3
Number Line
ο
−1
−4
0
−3
1
2
•
−2 −1
3
0
4
1
5
Interval Notation
x<3
( −∞, 3)
x ≥ −2
[ −2, ∞)
2<x<4
(2, 4)
1≤x≤3
[1, 3]
2
ο
ο
Inequality
−1
0
1
2
3
4
5
−2
−1
0
•1
2
•3
4
Example 2
Use interval notation to represent the set of
numbers.
4≤x<6
Solution
[4, 6)
4 is included, but 6 is not
Your Turn
Use interval notation to represent the sets of
numbers x ≤ 2 or 3 < x ≤ 11.
Solution
(−∞, 2] or (3, 11]
Another way to represent sets is set-builder notation.
Set-builder notation uses the properties of the
elements in the set to define the set. Inequalities
and the element symbol (∈) are often used in setbuilder notation. The set of striped-billiard-ball
numbers, or {9, 10, 11, 12, 13, 14, 15}, is represented
below in set-builder notation.
The set of all numbers x such that x has the given properties
{x І 8 < x ≤ 15 and x ∈ N}
Read the above as “the set of all numbers x such that x is greater
than 8 and less than or equal to 15 and x is a natural number.”
Some representations of the same sets of real
numbers are shown below.
Methods of Set Notation
Words
All real numbers
except 1
Positive odd
numbers
Numbers within 3
units of 2
Roster Notation
Interval Notation
Set-Builder Notation
Cannot be written
in roster notation
(−∞, 1) or (1, ∞)
{x І x ≠ 1}
{1, 3, 5, 7, …}
Cannot be written
in roster notation
Cannot be
notated using
interval notation
[−1, 5]
{x І x = 2n − 1 and
n ∈ N}
{x І −1 ≤ x ≤ 5}
Example 3
Rewrite each set in the indicated notation.
{ x І x = 2n and n ∈ N}; words
positive even numbers
−3 < x ≤ 5; interval notation
(−3, 5]
Your Turn
Rewrite each set in the indicated notation.
{2, 4, 6, 8}; words
even numbers between 1 and 9
[99, ∞); set-builder notation
{x І x ≥ 99}
Students may confuse (3, 5) used as interval
notation with (3, 5) used as an ordered pair.
Common symbols can be used in multiple
mathematical context. Therefore, it is
extremely important that you pay attention to
the context of the problem.
What are the similarities and differences
among roster, interval, and set-builder
notation?
Similarities
Differences
What are the advantages/disadvantages
of having multiple representations to
describe the same data set?
Advantages
Disadvantages
Essential Question
How can you tell if a number belongs
to the set of rational numbers?
If the number can be expressed as a
quotient (or ratio) of two integers,
where the denominator is not zero.
Q: Why do the other numbers refuse to take
2, 3, and 5 seriously?
A: They’re completely irrational.