Transcript Section 1.1

Math in Our World
Section 2.1
The Nature of Sets
Learning Objectives
 Define set.
 Write sets in three 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.
Three 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.
•Set-Builder Notation
– This method uses a variable, braces, and a
vertical bar | that is read as “such that.”
– A variable is a symbol (usually a letter) that can
represent different elements of a 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 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) Oregon  A, where A is the set of states
west of the Mississippi River.
(b) 27  {1, 5, 9, 13, 17, . . .}
(c) z  {v, w, x, y, z}
EXAMPLE 3
Understanding Set Notation
SOLUTION
(a) Oregon is west of the Mississippi, so Oregon is
an element of A. The statement is true.
(b) 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.
(c)
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.
Set-Builder Notation
Such
that
S =
Set
S
is
The
Set
of
{ x │ condition(s) }
All
elements
x
EXAMPLE 5
Writing a Set Using
Set-Builder Notation
Use set-builder notation to designate each set,
then write how your answer would be read
aloud.
(a) The set R contains the elements 2, 4, and 6.
(b) The set W contains the elements red,
yellow, and blue.
EXAMPLE 5
Writing a Set Using
Set-Builder Notation
SOLUTION
(a)R = {x │ x  E and x  7}
The set of all x such that x is an even natural
number and x is less than 7.
(b) W = {x │ x is a primary color}
The set of all x such that x is a primary color.
EXAMPLE 6
Using Different Set Notations
Designate the set S with elements 32, 33, 34, 35, …
using
(a) The roster method.
(b) The descriptive method.
(c) Set-builder notation.
SOLUTION
(a) {32, 33, 34, 35, . . .}
(b) The set S is the set of natural numbers greater than 31.
(c) {x │ x  N and x  31}
EXAMPLE 7
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 8
Classifying Sets as
Finite or Infinite
Classify each set as finite or infinite.
(a) {x │ x  N and x  100}
(b) Set R is the set of letters used to make Roman
numerals.
(c) {100, 102, 104, 106, . . .}
(d) Set M is the set of people in your immediate
family.
(e) Set S is the set of songs that can be written.
EXAMPLE 8
Classifying Sets as
Finite or Infinite
SOLUTION
(a) The set is finite since there are 99 natural numbers
that are less than 100.
(b) The set is finite since the letters used are C, D, I, L,
M, V, and X.
(c) The set is infinite since it consists of an unlimited
number of elements.
(d) The set is finite since there is a specific number of
people in your immediate family.
(e) 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 9
Identifying Empty Sets
Which of the following sets are empty?
(a) The set of woolly mammoth fossils in
museums.
(b) {x | x is a living woolly mammoth}
(c) {}
(d) {x | x is a natural number between 1 and 2}
EXAMPLE 9
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) Woolly mammoths have been extinct for almost
8,000 years, so this set is most definitely empty.
(c) Be careful! Each instance of { } and  represents the
empty set, but {} is a set with one element: .
(d) This set is empty because there are no natural
numbers between 1 and 2.
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 10
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 10
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 11
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 11
Deciding if Sets are
Equal or Equivalent
SOLUTION
(a) Equivalent
(b) Equal and equivalent
(c) Neither
(d) Equivalent
(e) Equal and equivalent
One-to-One Correspondence
Two sets have a one-to-one correspondence
of elements if each element in the first set can
be paired with exactly one element of the
second set and each element of the second
set can be paired with exactly one element of
the first set.
EXAMPLE 12
Putting Sets in One-to-One
Correspondence
Show that …
(a) the sets {8,16, 24, 32} and {s, t, u, v} have a
one-to-one correspondence and
(b) the sets {x, y, z} and {5, 10} do not have a
one-to-one correspondence.
EXAMPLE 12
Putting Sets in One-to-One
Correspondence
SOLUTION
(a) We need to demonstrate that each element of one
set can be paired with one and only one element of the
second set. One possible way to show a one-to-one
correspondence is this:
{8, 16, 24, 32}
{ s, t, u, v }
(b) The elements of the sets {x, y, z} and {5, 10} can’t be
put in one-to-one correspondence. No matter how we
try, there will be an element in the first set that doesn’t
correspond to any element in the second set.
Correspondence vs. Equivalence
Two sets are
• Equivalent if you can put their elements in
one-to-one correspondence.
• Not equivalent if you cannot put their
elements in one-to-one correspondence.