Transcript Set

2
THE NATURE OF
SETS
Copyright © Cengage Learning. All rights reserved.
2.1
Sets, Subsets, and Venn
Diagrams
Copyright © Cengage Learning. All rights reserved.
Denoting Sets
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Denoting Sets
“Set: a collection of objects.” What is a collection?
“Collection: an accumulation.” What is an accumulation?
“Accumulation: a collection, a pile, or a heap.”
We see that the word collection gives us a circular
definition. What is a pile?
“Pile: a heap.” What is a heap?
“Heap: a pile.”
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Denoting Sets
Sets are usually specified in one of two ways. The first is by
description, and the other is by the roster method.
In the description method, we specify the set by
describing it in such a way that we know exactly which
elements belong to it.
An example is the set of 50 states in the United States of
America. We say that this set is well defined.
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Denoting Sets
The distinctive property that determines the inclusion or
exclusion of a particular element is called the defining
property of the set.
In the roster method, the set is defined by listing the
members. The objects in a set are called members or
elements of the set and are said to belong to or be
contained in the set.
For example, instead of defining a set as the set of all
students in this class who received a C or better on the first
examination, we might simply define the set by listing
its members: {Howie, Mary, Larry}.
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Denoting Sets
Sets are usually denoted by capital letters, and the notation
used for sets is braces. Thus, the expression
A = {4, 5, 6}
means that A is the name for the set whose members are
the numbers 4, 5, and 6.
Sometimes we use braces with a defining property, as in
the following examples:
{states in the United States of America}
{all students in this class who received an A on the first
test}
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Denoting Sets
If S is a set, we write a  S if a is a member of the set S,
and we write b  S if b is not a member of the set S.
Thus, “a  ” means that the variable a is an integer,
and the statement “b  , b  0” means that the variable b
is a nonzero integer.
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Example 1 – Set member notation
Let C = cities in California, a = city of Anaheim,
and b = city of Berlin.
Use set membership notation to describe relations among
a, b, and C.
Solution:
a  C; b  C
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Denoting Sets
A common use of set terminology is to refer to certain sets
of numbers.
Some examples of set are listed below.
{1, 2, 3, 4, . . .}
Set of natural, or counting, numbers
{0, 1, 2, 3, 4, . . .}
Set of whole numbers
{. . . , –2, –1, 0, 1, 2, . . .}
Set of integers
Set of rational numbers
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Denoting Sets
A new notation called set-builder notation was invented to
allow us to combine both the roster and the description
methods.
Consider:
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Denoting Sets
We now use set-builder notation for the set of rational
numbers:
{ | a is an integer and b is a nonzero integer}
Read this as: “The set of all
b is a nonzero integer.”
such that a is an integer and
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Example 2 – Writing sets by roster
Specify the given sets by roster. If the set is not well
defined, say so.
a. {counting numbers between 10 and 20}
b. {x | x is an integer between –20 and 20}
Solution:
a. {11, 12, 13, 14, 15, 16, 17, 18, 19}
Notice that between does not include the first and last
numbers.
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Example 2 – Solution
cont’d
b. {–19, –18, –17, ...,17, 18, 19}
Notice that ellipses (three dots) are used to denote some
missing numbers.
When using ellipses, you must be careful to list enough
elements so that someone looking at the set can see the
intended pattern.
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Example 3 – Writing sets by description
Specify the given sets by description.
a. {1, 2, 3, 4, 5, ...}
b. {0, 1, 2, 3, 4, 5, ...}
c. {x | x  }
d. {12, 14, 16, ..., 98}
e. {4, 44, 444, 4444, . . .}
f. {m, a, t, h, e, i, c, s}
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Example 3 – Solution
Answers may vary.
a. Counting (or natural) numbers
b. Whole numbers
c. We would read this as “The set of all x such that x
belongs to the set of integers.” More simply, the answer
is integers.
d. {Even numbers between 10 and 100}
e. {Counting numbers whose digits consist of fours only}
f. {Distinct letters in the word mathematics}
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Equal and Equivalent Sets
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Equal and Equivalent Sets
We say that two sets A and B are equal, written A = B,
if the sets contain exactly the same elements.
Thus, if E = {2, 4, 6, 8, . . .}, then
{x | x is an even counting number} = {x | x  E}
The order in which you represent elements in a set has no
effect on set membership. Thus,
{1, 2, 3} = {3, 1, 2} = {2, 1, 3} = . . .
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Equal and Equivalent Sets
Also, if an element appears in a set more than once, it is
not generally listed more than a single time. For example,
{1, 2, 3, 3} = {1, 2, 3}
Another possible relationship between sets is that of
equivalence.
Two sets A and B are equivalent, written A B, if they
have the same number of elements. Equivalent sets do not
need to be equal sets, but equal sets are always
equivalent.
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Example 4 – Equal and equivalent sets
Which of the following sets are equivalent? Are any equal?
Solution:
All of the given sets are equivalent. Notice that no two of
them are equal, but they all share the property of
“fourness.”
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Equal and Equivalent Sets
The number of elements in a set is often called its
cardinality. The cardinality of the sets in Example 4 is 4;
that is, the common property of the sets is the cardinal
number of the set.
The cardinality of a set S is denoted by |S|.
Equivalent sets with four elements each have in common
the property of “fourness,” and thus we would say that
their cardinality is 4.
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Example 5 – Finding cardinality
Find the cardinality of each of the following sets.
a.
b. S = { }
c. T = {states of the United States}
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Example 5 – Solution
a. The cardinality of R is 4, so we write |R| = 4.
b. The cardinality of S (the empty set) is 0, so we write
|S| = 0.
c. The cardinality of T is 50, or |T| = 50.
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Universal and Empty Sets
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Universal and Empty Sets
We now consider two important sets in set theory. The first
is the set that contains every element under consideration,
and the second is the set that contains no elements.
A universal set, denoted by U, contains all the elements
and the empty set contains no elements, and thus has
cardinality 0.
The empty set is denoted by { } or
Do not confuse the notations
.
, 0, and { }.
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Universal and Empty Sets
The symbol denotes a set with no elements; the symbol
0 denotes a number; and the symbol { } is a set with one
element (namely, the set containing )
For example, if U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, then all sets we
would be considering would have elements only among the
elements of U. No set could contain the number 10, since
10 is not in that agreed-upon universe.
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Universal and Empty Sets
A universal set must be specified or implied, and it must
remain fixed for that problem. However, when a new
problem is begun, a new universal set can be specified.
The following are examples of descriptions of the
empty set:
{living saber-toothed tigers}
{counting numbers less than 1}
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Venn Diagrams
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Venn Diagrams
A set is a collection, and a useful way to depict a set is to
draw a circle or an oval as a representation for the set.
The elements are depicted inside the circle, and objects not
in the set are shown outside the circle.
The universal set contains all the elements under
consideration in a given discussion and is depicted as a
rectangle.
This representation of a set is called a Venn diagram,
after John Venn (1834–1923).
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Example 6 – Venn diagram for a given set
Let the universal set be all of the cards in a deck of cards.
Draw a Venn diagram for the set of hearts.
Solution:
It is customary to represent the universal set as a rectangle
(labeled U) and the set of hearts (labeled H) as a circle, as
shown in Figure 2.1.
Venn diagram for a deck of cards
Figure 2.1
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Example 6 – Solution
cont’d
Note that
In the Venn diagram, the sets involved are too large to list
all of the elements individually in either H or U, but we can
say that the two of hearts (labeled ) is a member of H,
whereas the two of diamonds (labeled ) is not a member
of H.
We write
 H,
whereas
H
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Venn Diagrams
The set of elements that are not in H is referred to as the
complement of H, and this is written using an overbar.
In Example 6, H = {spades, diamonds, clubs}.
A Venn diagram for complement is shown in Figure 2.2.
Venn diagrams for a set and for its complement
Figure 2.2
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Venn Diagrams
Notice that any set S divides the universe into two regions
as shown in Figure 2.3.
General representation of a set S
Figure 2.3
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Venn Diagrams
Notice that the cardinality of the deck of cards in Example 6
is 52, and the cardinality of H is 13.
Since the deck of cards is the universal set for Example 6,
we can symbolize the cardinality of these sets as follows:
|H| = 13
and
|U| = 52
Note that H  |H|.
In words, a set is not the same as its cardinality.
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Subsets and Proper Subsets
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Subsets and Proper Subsets
Most applications will involve more than one set, so we
begin by considering the relationships between two
sets A and B. The various possible relationships are shown
in Figure 2.4.
Relationships between two sets A and B
Figure 2.4
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Subsets and Proper Subsets
We say that A is a subset of B, which in set theory is
written A  B, if every element of A is also an element of B
(see Figure 2.4a).
Similarly, B  A if every element of B is also an element
of A (Figure 2.4b).
Figure 2.4c shows two equal sets.
Finally, A and B are disjoint if they have no elements in
common (Figure 2.4d).
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Example 8 – Subsets of a given set
Find all possible subsets of C = {5, 7}.
Solution:
{5}, {7} are obvious subsets.
{5, 7} is also a subset, since both 5 and 7 are elements of C.
{ } is also a subset of C. It is a subset because all of its
elements belong to C.
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Example 8 – Solution
cont’d
Stated a different way, if it were not a subset of C, we
would have to be able to find an element of { } that is
not in C.
Since we cannot find such an element, we say that the
empty set is a subset of C (and, in fact, the empty set is a
subset of every set).
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Subsets and Proper Subsets
The subsets of C can be classified into two categories:
proper and improper. Since every set is a subset of itself,
we immediately know one subset for any given set: the set
itself.
A proper subset is a subset that is not equal to the original
set; that is, A is a proper subset of a subset B, written
A  B, if A is a subset of B and A ≠ B. An improper subset
of a set A is the set A.
We see there are three proper subsets of C = {5,7} :
and {7}. There is one improper subset of C: {5, 7}.
, {5}
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Example 9 – Classifying proper and improper subsets
Find the proper and improper subsets of A = {2, 4, 6, 8}.
What is the cardinality of A?
Solution:
The cardinality of A is 4 (because there are 4 elements
in A). There is one improper subset: {2, 4, 6, 8}. The proper
subsets are as follows:
{ },
{2}, {4}, {6}, {8},
{2, 4}, {2, 6}, {2, 8}, {4, 6}, {4, 8}, {6, 8},
{2, 4, 6}, {2, 4, 8}, {2, 6, 8}, {4, 6, 8}
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Subsets and Proper Subsets
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Subsets and Proper Subsets
Sometimes we are given two sets X and Y, and we know
nothing about the way they are related. In this situation, we
draw a general figure, such as the one shown in Figure 2.5.
General Venn diagram for two sets
Figure 2.5
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Example 10 – Regions in a Venn diagram
Name the regions in Figure 2.6 described by each of the
following.
a. A
b. C
c. A
d. B
e. A  B
f. A and C are disjoint
General Venn diagram for three sets
Figure 2.6
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Example 10 – Solution
a. A is regions I, IV, V, and VII.
b. C is regions III, IV, VI, and VII.
c. A is regions II, III, VI, and VIII.
d. B is regions I, III, IV, and VIII.
e. A  B means that regions I and IV are empty.
f. A and C are disjoint means that regions IV and VII are
empty.
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