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Math in Our World
Section 2.1
The Nature of Sets
Learning Objectives
 Define set.
 Write sets in two different ways.
 Classify sets as finite or infinite.
 Define the empty set.
 Find the cardinality of a set.
 Decide if two sets are equal or equivalent.
Sets
A set is a well-defined collection of objects.
By well-defined we mean that given any object, we
can definitely decide whether it is or is not in the
set.
Each object in a set is called an element or a
member of the set.
EXAMPLE 1
Listing the Elements in a Set
Write the set of months of the year that begin
with the letter M.
SOLUTION
The months that begin with M are March and May. So, the
answer can be written in set notation as
M = {March, May}
Each element in the set is listed within braces and is
separated by a comma.
Two methods are commonly
used to designate a set:
•Roster Form
– The elements of the set are listed between
braces, with commas between the elements.
•Description
– This uses a short statement to describe the set.
Sets are generally named with a
capital letter.
•The Set of Natural Numbers (Counting
Numbers) is listed:
N = {1, 2, 3, 4 …}
•The Integers:
I = {…-3, -2, -1, 0, 1, 2, 3…}
The three dots, or an
Ellipsis, indicates that the
pattern continues
indefinitely.
EXAMPLE 2
Writing Sets Using the
Roster Method
Use the roster method to do the following:
(a) Write the set of natural numbers less than 6.
(b) Write the set of natural numbers greater than 4.
SOLUTION
(a) {1, 2, 3, 4, 5}
(b) {5, 6, 7, 8, . . .}
Set Notation
The symbol  is used to show that an object is a
member or element of a set. For example, let set
A = {2, 3, 5, 7, 11}.
Since 2 is a member of set A, it can be written as
2  {2, 3, 5, 7, 11} or 2  A
Likewise, 5  {2, 3, 5, 7, 11} or 5  A
When an object is not a member of a set, the
symbol  is used. Because 4 is not an element of
set A, this fact is written as
4  {2, 3, 5, 7, 11} or 4  A
EXAMPLE 3
Understanding Set Notation
Decide whether each statement is true or false.
(a) 27  {1, 5, 9, 13, 17, . . .}
(b) z  {v, w, x, y, z}
EXAMPLE 3
Understanding Set Notation
SOLUTION
(a) The pattern shows that each element is 4 more
than the previous element. So the next three
elements are 21, 25, and 29; this shows that 27 is
not in the set. The statement is false.
(b) The letter z is an element of the set, so the
statement is false.
EXAMPLE 4
Describing a Set in Words
Use the descriptive method to describe the set
E containing 2, 4, 6, 8, . . . .
SOLUTION
The elements in the set are called the even natural
numbers. The set E is the set of even natural numbers.
EXAMPLE 5
Using Different Set Notations
Designate the set S with elements 32, 33, 34, 35, …
using
(a) The roster method.
(b) The descriptive method.
SOLUTION
(a) {32, 33, 34, 35, . . .}
(b) The set S is the set of natural numbers greater than 31
EXAMPLE 6
Writing a Set Using an Ellipsis
Using the roster method, write the set containing
all even natural numbers between 99 and 201.
SOLUTION
{100, 102, 104, . . . , 198, 200}
If a set contains many elements, we can again use an
ellipsis to represent the missing elements.
Finite and Infinite Sets
If a set has no elements or a specific
natural number of elements, then it is
called a finite set. A set that is not a finite
set is called an infinite set.
The set {p, q, r, s} is called an finite set since it has
four members: p, q, r, and s. The set {10, 20, 30, . . .}
is called an infinite set since it has an unlimited
number of elements: the natural numbers that are
multiples of 10.
EXAMPLE 7
Classifying Sets as
Finite or Infinite
Classify each set as finite or infinite.
(a) Set R is the set of letters used to make Roman
numerals.
(b) {100, 102, 104, 106, . . .}
(c) Set M is the set of people in your immediate
family.
(d) Set S is the set of songs that can be written.
EXAMPLE 7
Classifying Sets as
Finite or Infinite
SOLUTION
(a) The set is finite since the letters used are C, D, I, L,
M, V, and X.
(b) The set is infinite since it consists of an unlimited
number of elements.
(c) The set is finite since there is a specific number of
people in your immediate family.
(d) The set is infinite because an unlimited number of
songs can be written.
Empty Set or Null Set
A set with no elements is called an empty
set or null set. The symbols used to
represent the null set are { } or .
EXAMPLE 8
Identifying Empty Sets
Which of the following sets are empty?
(a) The set of woolly mammoth fossils in
museums.
(b) {}
EXAMPLE 8
Identifying Empty Sets
SOLUTION
(a) There is certainly at least one woolly mammoth
fossil in a museum somewhere, so the set is not empty..
(b) Be careful! Each instance of { } and  represents
the empty set, but {} is a set with one element: .
Cardinal Number
The cardinal number of a finite set is the
number of elements in the set. For a set A
the symbol for the cardinality is n(A),
which is read as “n of A.”
For example, the set R = {2, 4, 6, 8, 10} has a cardinal
number of 5 since it has 5 elements. This could also
be stated by saying the cardinality of set R is 5.
EXAMPLE 9
Finding the Cardinality of a Set
Find the cardinal number of each set.
(a) A = {5, 10, 15, 20, 25, 30}
(b) B = {10, 12, 14, . . . , 28, 30}
(c) C = {16}
(d) 
EXAMPLE 9
Finding the Cardinality of a Set
SOLUTION
(a) n(A) = 6 since set A has 6 elements
(b) n(B) = 11 since set B has 11 elements
(c) n(C) = 1 since set C has 1 element
(d) n() = 0 since there are no elements in an empty set
Equal and Equivalent Sets
Two sets A and B are equal (written A = B)
if they have exactly the same members or
elements. Two finite sets A and B are said
to be equivalent (written A  B) if they
have the same number of elements: that
is, n(A) = n(B).
EXAMPLE 10
Deciding if Sets are
Equal or Equivalent
State whether each pair of sets is equal,
equivalent, or neither.
(a) {p, q, r, s}; {a, b, c, d}
(b) {8, 10, 12}; {12, 8, 10}
(c) {213}; {2, 1, 3}
(d) {1, 2, 10, 20}; {2, 1, 20, 11}
(e) {even natural numbers less than 10};
{2, 4, 6, 8}
EXAMPLE 10
Deciding if Sets are
Equal or Equivalent
SOLUTION
(a) Equivalent
(b) Equal and equivalent
(c) Neither
(d) Equivalent
(e) Equal and equivalent
Classwork
p. 50-53: 7, 9, 11, 15, 17, 21, 27, 35, 37, 49, 51, 53,
57, 59, 53, 65, 67, 75, 79, 80, 83, 85, 87, 91 (a, b),
96 (a, b), 98 (a, b)