4. 4
Sets
Any group or collection of objects is called a set. The
objects that belong in a set are the elements, or members,
of the set. For example, the set consisting of the four
seasons has spring, summer, fall, and winter as its
elements.
The following two methods are often used to designate a
set.
• Describe the set using words.
• List the elements of the set inside a pair of braces, { }.
This method is called the roster method. Commas are
used to separate the elements.
5. 5
Sets
For instance, let’s use S to represent the set consisting of
the four seasons. Using the roster method, we would write
S = {spring, summer, fall, winter}
The order in which the elements of a set are listed is not
important. Thus the set consisting of the four seasons can
also be written as
S = {winter, spring, fall, summer}
6. 6
Sets
The following table gives two examples of sets, where each
set is designated by a word description and also by using
the roster method.
Define Sets by Using a Word Description and the Roster Method
Table 2.1
7. 7
Example 1 – Use the Roster Method to Represent a Set
Use the roster method to represent the set of the days in a
week.
Solution:
8. 8
Example 2 – Use a Word Description to Represent a Set
Write a word description for the set
Solution:
Set A is the set of letters of the English alphabet.
10. 10
Sets
The set of natural numbers is also called the set of
counting numbers. The three dots ... are called an ellipsis
and indicate that the elements of the set continue in a
manner suggested by the elements that are listed.
The integers ... , –4, –3, –2, –1 are negative integers. The
integers 1, 2, 3, 4, ... are positive integers. Note that the
natural numbers and the positive integers are the same set
of numbers. The integer zero is neither a positive nor a
negative integer.
11. 11
Sets
If a number in decimal form terminates or repeats a block
of digits without end, then the number is a rational number.
Rational numbers can also be written in the form
where p and q are integers and q 0. For example,
are rational numbers.
12. 12
Sets
The bar over the 27 means that the block of digits 27
repeats without end; that is,
A decimal that neither terminates nor repeats is an
irrational number. For instance, 0.35335333533335. . .
is a nonterminating, nonrepeating decimal and thus is
an irrational number.
Every real number is either a rational number or an
irrational number.
13. 13
Example 3 – Use the Roster Method to Represent a Set of Numbers
Use the roster method to write each of the given sets.
a. The set of natural numbers less than 5
b. The solution set of
c. The set of negative integers greater than –4
Solution:
a. The set of natural numbers is given by
{1, 2, 3, 4, 5, 6, 7, ...}. The natural numbers less than 5
are 1, 2, 3, and 4. Using the roster method, we write this
set as {1, 2, 3, 4}.
14. 14
Example 3 – Solution
b. Adding –5 to each side of the equation produces x = –6.
The solution set of
c. The set of negative integers greater than –4 is
{–3, –2, –1}.
cont’d
16. 16
Definitions Regarding Sets
A set is well defined if it is possible to determine whether
any given item is an element of the set. For instance, the
set of letters of the English alphabet is well defined. The set
of great songs is not a well-defined set.
It is not possible to determine whether any given song is an
element of the set or is not an element of the set because
there is no standard method for making such a judgment.
17. 17
Definitions Regarding Sets
The statement “4 is an element of the set of natural
numbers” can be written using mathematical notation
as 4 N. The symbol is read “is an element of.”
To state that “–3 is not an element of the set of natural
numbers,” we use the “is not an element of” symbol, , and
write –3 N.
18. 18
Example 4 – Apply Definitions Regarding Sets
Determine whether each statement is true or false.
Solution:
a. Since 4 is an element of the given set, the statement is
true.
b. There are no negative natural numbers, so the
statement is false.
19. 19
Example 4 – Solution
c. Since is not an integer, the statement is true.
d. The word nice is not precise, so the statement is false.
cont’d
20. 20
Definitions Regarding Sets
The empty set, or null set, is the set that contains no
elements. The symbol or { } is used to represent the
empty set. As an example of the empty set, consider the
set of natural numbers that are negative integers.
Another method of representing a set is set-builder
notation. Set-builder notation is especially useful when
describing infinite sets.
21. 21
Definitions Regarding Sets
For instance, in set-builder notation, the set of natural
numbers greater than 7 is written as follows:
The preceding set-builder notation is read as “the set of all
elements x such that x is an element of the set of natural
numbers and x is greater than 7.” It is impossible to list all
the elements of the set, but set-builder notation defines the
set by describing its elements.
22. 22
Example 5 – Use Set-Builder Notation to Represent a Set
Use set-builder notation to write the following sets.
a. The set of integers greater than –3
b. The set of whole numbers less than 1000
Solution:
23. 23
Definitions Regarding Sets
A set is finite if the number of elements in the set is a
whole number.
The cardinal number of a finite set is the number of
elements in the set. The cardinal number of a finite set A is
denoted by the notation n(A).
For instance, if A = {1, 4, 6, 9}, then n(A) = 4. In this case,
A has a cardinal number of 4, which is sometimes stated as
“A has a cardinality of 4.”
24. 24
Example 6 – The Cardinality of a Finite Set
Find the cardinality of each of the following sets.
Solution:
a. Set J contains exactly two elements, so J has a
cardinality of 2. Using mathematical notation, we state
this as n(J) = 2.
b. Only a few elements are actually listed. The number of
natural numbers from 1 to 31 is 31. If we omit the
numbers 1 and 2, then the number of natural numbers
from 3 to 31 must be 31 – 2 = 29. Thus n(S) = 29.
25. 25
Example 6 – Solution
c. Elements that are listed more than once are counted
only once. Thus n(T) = 3.
cont’d
26. 26
Definitions Regarding Sets
The following definitions play an important role in our work
with sets.
For instance {d, e, f } = {e, f, d }.
27. 27
Example 7 – Equal Sets and Equivalent Sets
State whether each of the following pairs of sets are equal,
equivalent, both, or neither.
Solution:
a. The sets are not equal. However, each set has exactly
five elements, so the sets are equivalent.
b. The first set has three elements and the second set has
four elements, so the sets are not equal and are not
equivalent.
