Transcript Set-Builder Notation

Set-Builder, Roster Notation,
and Classifying Numbers
MATH 017
Intermediate Algebra
S. Rook
Overview
• Section 1.2, Objective 2 in the textbook
– Set-Builder & Roster Notation
– Classifying Numbers
2
Set-Builder & Roster Notation
Set-Builder Notation
• Set-Builder Notation: describes, but
does not explicitly list the elements of a
set.
• Example: {x | x is an even number},
– The | (vertical bar) is pronounced “such that”
• A common exercise is to take a set written
in set-builder notation and convert it into
what is known as roster notation.
4
Roster Notation
• Roster Notation: explicitly listing the
elements of a set.
– When listing elements, we use set notation
and place the elements between and left {
and right } (called curly braces).
5
Set-Builder & Roster Notation
(Example)
Ex 1: Given the set {x | x is an even
number between 0 and 10 inclusive}, list
its members using roster notation.
6
Set-Builder & Roster Notation
(Example)
Ex 2: Given the set {x | x is a factor of 5},
list its members using roster notation.
7
Set-Builder & Roster Notation
(Example 2 Continued)
We use … (ellipses) to denote a set
extending infinitely in the same pattern.
The set of even numbers can then be
expressed as {0, 2, 4, 6, …}
The set of odd numbers can then be
expressed as {1, 3, 5, 7, 9, 11,…}
8
Set-Builder & Roster Notation
(Example)
Ex 3: Given the set
{x | x is both an odd and even number},
list its members using roster notation.
9
Classifying Numbers
Classifying Numbers
• Sets of numbers to be familiar with:
– Natural numbers (counting numbers):
{1, 2, 3, 4, 5,…}
– Whole numbers: the natural numbers along
with 0. {0, 1, 2, 3, 4,…}
– Integers: the natural numbers, opposite of
the natural numbers, and zero.
{…, -2, -1, 0, 1, 2,…}
11
Classifying Numbers (continued)
– Rational numbers: any number that can be
expressed as the quotient of two integers, a,
b, b ≠ 0. {a/b | a and b are integers, b ≠ 0}
– Irrational numbers: any number that
CANNOT be expressed as the quotient of two
integers.
– Real numbers: any number that lies on the
number line.
12
Classifying Numbers (continued)
• A number can lie in more than one set.
– The chart on page 12 in the textbook is helpful.
• Presented in a hierarchical manner.
• As the chart is traced upwards, the number sets proceed
from the specific to the general.
– Once a number has been classified as specifically as
possible, follow the chart upwards until the real
numbers are reached.
• Example: 7 can most specifically be classified as a natural
number. Following the chart upwards, it can be a whole
number, integer, rational number, and a real number.
13
Classifying Numbers (continued)
14
Classifying Numbers (Example)
Ex 4: Which sets of numbers does -0.666…
belong?
15
Classifying Numbers (Example 4
Continued)
• As shown in the example, numbers can be in
hidden format. A few to look out for:
– Terminating or repeating decimals: are equivalent
to fractions hence rational numbers
– Square roots: are equivalent to natural numbers if
the number under the radical is a perfect square, a
whole number if the number under the radical is 0,
and an irrational number otherwise.
• A common exercise is to be presented with a set
of numbers and then asked to classify each
member of the set.
16
Classifying Numbers (Example)
Ex 5: Given the set S   7, 16 ,0, 1 , 3 ,  ,0.3,0.3 ,



2

list the elements in the following sets:
a)
b)
c)
d)
e)
f)
Irrational numbers
Whole numbers
Integers
Natural numbers
Rational numbers
Real numbers
17
Writing Set-Builder Notation
Ex 6: Given the set S = {-3, -2, -1, 0, 1, 2},
express S in set-builder notation – be
specific as possible.
18
Writing Set-Builder Notation
(Continued)
Ex 7: Given the set S = {1, 2, 3, 4, 5},
express S in set-builder notation – be
specific as possible.
19
Summary
• After studying these slides, you should know how to do
the following:
– Understand the concepts of roster & set-builder notation
– Convert a set expressed in set-builder to roster notation and vice
versa
– Distinguish between natural numbers, whole numbers, integers,
rational numbers, and real numbers
– Given a random set of numbers, identify the elements that
correspond with at least one classification (see above)
• Additional Practice
– Attempt the suggested problems found on the course website.
20