2.3 Set Operations and Cartesian Products
2.3
63
Set Operations and Cartesian Products
Intersection of Sets Two candidates, Adelaide Boettner and David Berman,
are running for a seat on the city council. A voter deciding for whom she should vote
recalled the following campaign promises made by the candidates. Each promise
is given a code letter.
t
s
m
p
c
FIGURE 4
Honest Adelaide Boettner
Determined David Berman
Spend less money, m
Emphasize traffic law enforcement, t
Increase service to suburban areas, s
Spend less money, m
Crack down on crooked politicians, p
Increase service to the city, c
The only promise common to both candidates is promise m, to spend less money.
Suppose we take each candidate’s promises to be a set. The promises of Boettner
give the set m, t, s, while the promises of Berman give m, p, c. The only element
common to both sets is m; this element belongs to the intersection of the two sets
m, t, s and m, p, c, as shown in color in the Venn diagram in Figure 4. In symbols,
m, t, s m, p, c m,
where the cap-shaped symbol represents intersection. Notice that the intersection
of two sets is itself a set.
Intersection of Sets
The intersection of sets A and B, written A B, is the set of elements
common to both A and B, or
A B {x| x A and x B}.
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Form the intersection of sets A and B by taking all the elements included in both sets,
as shown in color in Figure 5.
A
B
EXAMPLE
U
A
1
Find the intersection of the given sets.
(a) 3, 4, 5, 6, 7 and 4, 6, 8, 10
Since the elements common to both sets are 4 and 6,
B
FIGURE 5
3, 4, 5, 6, 7 4, 6, 8, 10 4, 6.
(b) 9, 14, 25, 30 and 10, 17, 19, 38, 52
These two sets have no elements in common, so
9, 14, 25, 30 10, 17, 19, 38, 52 0.
(c) 5, 9, 11 and 0
There are no elements in 0, so there can be no elements belonging to both
5, 9, 11 and 0. Because of this,
White light can be viewed
as the intersection of the
three primary colors.
5, 9, 11 0 0.
Examples 1(b) and 1(c) show two sets that have no elements in common. Sets
with no elements in common are called disjoint sets. A set of dogs and a set of
cats would be disjoint sets. In mathematical language, sets A and B are disjoint if
A B 0. Two disjoint sets A and B are shown in Figure 6.
Union of Sets
A
B
U
At the beginning of this section, we showed lists of campaign promises of two candidates running for city council. Suppose a pollster
wants to summarize the types of promises made by candidates for the office.
The pollster would need to study all the promises made by either candidate, or
the set
Disjoint sets
m, t, s, p, c,
FIGURE 6
the union of the sets of promises made by the two candidates, as shown in color in
the Venn diagram in Figure 7. In symbols,
t
s
m
p
c
m, t, s m, p, c m, t, s, p, c,
where the cup-shaped symbol denotes set union. Be careful not to confuse this
symbol with the universal set U. Again, the union of two sets is a set.
FIGURE 7
Union of Sets
The union of sets A and B, written A B, is the set of all elements
belonging to either of the sets, or
A B {x| x A or x B}.
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A
Set Operations and Cartesian Products
65
Form the union of sets A and B by taking all the elements of set A and then
including the elements of set B that are not already listed, as shown in color in
Figure 8.
B
EXAMPLE
U
AB
FIGURE 8
2
Find the union of the given sets.
(a) 2, 4, 6 and 4, 6, 8, 10, 12
Start by listing all the elements from the first set, 2, 4, and 6. Then list all the
elements from the second set that are not in the first set, 8, 10, and 12. The union
is made up of all these elements, or
2, 4, 6 4, 6, 8, 10, 12 2, 4, 6, 8, 10, 12.
(b) a, b, d, f, g, h and c, f, g, h, k
The union of these sets is
a, b, d, f, g, h c, f, g, h, k a, b, c, d, f, g, h, k.
(c) 3, 4, 5 and 0
Since there are no elements in 0, the union of 3, 4, 5 and 0 contains only the
elements 3, 4, and 5, or
3, 4, 5 0 3, 4, 5.
FOR FURTHER THOUGHT
The arithmetic operations of addition and
multiplication, when applied to numbers, have
some familiar properties. If a, b, and c are real
numbers, then the commutative property of
addition says that the order of the numbers
being added makes no difference: a b b a. (Is there a commutative property of
multiplication?) The associative property of
addition says that when three numbers are
added, the grouping used makes no difference:
(a b) c a (b c). (Is there an
associative property of multiplication?) The
number 0 is called the identity element for
addition since adding it to any number does
not change that number: a 0 a. (What
is the identity element for multiplication?)
Finally, the distributive property of multiplication over addition says that a(b c) ab ac.
(Is there a distributive property of addition over
multiplication?)
For Group Discussion
Now consider the operations of union and
intersection, applied to sets. By recalling
definitions, or by trying examples, answer the
following questions.
1. Is set union commutative? How about set
intersection?
2. Is set union associative? How about set
intersection?
3. Is there an identity element for set union? If
so, what is it? How about set intersection?
4. Is set intersection distributive over set
union? Is set union distributive over set
intersection?
Recall from the previous section that A represents the complement of set A. Set
A is formed by taking all the elements of the universal set U that are not in A.
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EXAMPLE
3
Let U 1, 2, 3, 4, 5, 6, 9,
A 1, 2, 3, 4,
B 2, 4, 6,
C 1, 3, 6, 9.
Find each set.
(a) A B
First identify the elements of set A, the elements of U that are not in set A:
A 5, 6, 9.
Now find A B, the set of elements belonging both to A and to B:
A B 5, 6, 9 2, 4, 6 6.
(b) B C 1, 3, 5, 9 2, 4, 5 1, 2, 3, 4, 5, 9.
(c) A B C
First find the set inside the parentheses:
B C 2, 4, 6 2, 4, 5 2, 4, 5, 6.
Now, find the intersection of this set with A.
A B C A 2, 4, 5, 6
1, 2, 3, 4 2, 4, 5, 6
2, 4
(d) A C B
Set A 5, 6, 9 and set C 2, 4, 5, with
A C 5, 6, 9 2, 4, 5 2, 4, 5, 6, 9.
Set B is 1, 3, 5, 9, so
A C B 2, 4, 5, 6, 9 1, 3, 5, 9 5, 9.
It is often said that mathematics is a “language.” As such, it has the advantage
of concise symbolism. For example, the set A B C is less easily expressed in
words. One attempt is the following: “The set of all elements that are not in both
A and B, or are in C.” The key words here (not, and, or) will be treated more
thoroughly in the chapter on logic.
EXAMPLE
4
Describe each of the following sets in words.
(a) A B C
This set might be described as “the set of all elements that are in A, and are in
B or not in C.”
(b) A C B
One possibility is “the set of all elements that are not in A or not in C, and are
not in B.”
Difference of Sets
We now consider the difference of two sets. Suppose that
A 1, 2, 3, . . . , 10 and B 2, 4, 6, 8, 10. If the elements of B are excluded (or
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67
taken away) from A, the set C 1, 3, 5, 7, 9 is obtained. C is called the difference
of sets A and B.
Difference of Sets
The difference of sets A and B, written A B, is the set of all elements
belonging to set A and not to set B, or
A
A B {x| x A and x B .
B
U
AB
FIGURE 9
Since x B has the same meaning as x B, the set difference A B can also be
described as x x A and x B, or A B. Figure 9 illustrates the idea of set
difference. The region in color represents A B.
EXAMPLE
5
Let U 1, 2, 3, 4, 5, 6, 7,
A 1, 2, 3, 4, 5, 6,
B 2, 3, 6,
C 3, 5, 7.
Find each set.
(a) A B
Begin with set A and exclude any elements found also in set B. So,
A B 1, 2, 3, 4, 5, 6 2, 3, 6 1, 4, 5.
(b) B A
To be in B A, an element must be in set B and not in set A. But all elements
of B are also in A. Thus, B A 0.
(c) A B C
From part (a), A B 1, 4, 5. Also, C 1, 2, 4, 6, so
A B C 1, 2, 4, 5, 6.
The results in Examples 5(a) and 5(b) illustrate that, in general,
A B B A.
Ordered Pairs When writing a set that contains several elements, the order in
which the elements appear is not relevant. For example, 1, 5 5, 1. However,
there are many instances in mathematics where, when two objects are paired, the
order in which the objects are written is important. This leads to the idea of the ordered pair. When writing ordered pairs, use parentheses (as opposed to braces,
which are reserved for writing sets).
Ordered Pairs
In the ordered pair a, b, a is called the first component and b is
called the second component. In general, a, b b, a.
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Two ordered pairs a, b and c, d are equal provided that their first components
are equal and their second components are equal; that is, a, b c, d if and only
if a c and b d.
EXAMPLE
6
Decide whether each statement is true or false.
(a) 3, 4 5 2, 1 3
Since 3 5 2 and 4 1 3, the ordered pairs are equal. The statement is
true.
(b) 3, 4 4, 3
Since these are sets and not ordered pairs, the order in which the elements are
listed is not important. Since these sets are equal, the statement is false.
(c) 7, 4 4, 7
These ordered pairs are not equal since they do not satisfy the requirements for
equality of ordered pairs. The statement is false.
Cartesian Product of Sets
A set may contain ordered pairs as elements. If
A and B are sets, then each element of A can be paired with each element of B, and
the results can be written as ordered pairs. The set of all such ordered pairs is called
the Cartesian product of A and B, written A B and read “A cross B.” The name
comes from that of the French mathematician René Descartes.
Cartesian Product of Sets
The Cartesian product of sets A and B, written A B, is
A B {(a, b)|a A and b B}.
EXAMPLE
7
Let A 1, 5, 9 and B 6, 7. Find each set.
(a) A B
Pair each element of A with each element of B. Write the results as ordered
pairs, with the element of A written first and the element of B written second.
Write as a set.
A B 1, 6, 1, 7, 5, 6, 5, 7, 9, 6, 9, 7
(b) B A
Since B is listed first, this set will consist of ordered pairs that have their components interchanged when compared to those in part (a).
B A 6, 1, 7, 1, 6, 5, 7, 5, 6, 9, 7, 9
It should be noted that the order in which the ordered pairs themselves are listed
is not important. For example, another way to write B A in Example 7 would be
6, 1, 6, 5, 6, 9, 7, 1, 7, 5, 7, 9.
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EXAMPLE 8
Let A 1, 2, 3, 4, 5, 6. Find A A.
In this example we take the Cartesian product of a set with itself. By pairing 1
with each element in the set, 2 with each element, and so on, we obtain the following set:
A A 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6,
2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6,
3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6,
4, 1, 4, 2, 4, 3, 4, 4, 4, 5, 4, 6,
5, 1, 5, 2, 5, 3, 5, 4, 5, 5, 5, 6,
6, 1, 6, 2, 6, 3, 6, 4, 6, 5, 6, 6.
It is not unusual to take the Cartesian product of a set with itself, as in Example 8. In fact, the Cartesian product in Example 8 represents all possible results
that are obtained when two distinguishable dice are rolled. Determining this Cartesian product is important when studying certain problems in counting techniques
and probability, as we shall see in later chapters.
From Example 7 it can be seen that, in general, A B B A, since
they do not contain exactly the same ordered pairs. However, each set contains
the same number of elements, six. Furthermore, nA 3, nB 2, and
nA B nB A 6. Since 3 2 6, you might conclude that the cardinal
number of the Cartesian product of two sets is equal to the product of the cardinal
numbers of the sets. In general, this conclusion is correct.
Cardinal Number of a Cartesian Product
If nA a and nB b, then n(A B) n(B A) n(A) n(B) n(B) n(A) ab.
EXAMPLE
9
Find nA B and nB A from the given information.
(a) A a, b, c, d, e, f, g and B 2, 4, 6
Since nA 7 and nB 3, nA B and nB A are both equal to 7 3,
or 21.
(b) nA 24 and nB 5
nA B nB A 24 5 120
Operations on Sets
Finding intersections, unions, differences, Cartesian
products, and complements of sets are examples of set operations. An operation is
a rule or procedure by which one or more objects are used to obtain another object.
The objects involved in an operation are usually sets or numbers. The most common
operations on numbers are addition, subtraction, multiplication, and division. For example, starting with the numbers 5 and 7, the addition operation would produce the
number 5 7 12. The multiplication operation would produce 5 7 35.
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The most common operations on sets are summarized below, along with their
Venn diagrams.
Set Operations
Let A and B be any sets, with U the universal set.
The complement of A, written A, is
A′
A x x U and x A.
A
U
The intersection of A and B is
A B x x A and x B.
A
B
A
B
A
B
U
A′
A
The union of A and B is
A B x x A or x B.
U
U
The difference of A and B is
FIGURE 10
A B x x A and x B.
The Cartesian product of A and B is
U
A B x, y x A and y B.
1
A
B
3
2
4
U
Numbering is arbitrary. The
numbers indicate four
regions, not cardinal
numbers.
FIGURE 11
Venn Diagrams
When dealing with a single set, we can use a Venn diagram as
seen in Figure 10. The universal set U is divided into two regions, one representing
set A and the other representing set A.
Two sets A and B within the universal set suggest a Venn diagram as seen in Figure 11, where the four resulting regions have been numbered to provide a convenient way to refer to them. (The numbering is arbitrary.) Region 1 includes those
elements outside of both set A and set B. Region 2 includes the elements belonging
to A but not to B. Region 3 includes those elements belonging to both A and B. How
would you describe the elements of region 4?
1
A
2
B
3
4
U
FIGURE 12
1
A
B
3
2
4
U
FIGURE 13
E X A M P L E 10 Draw a Venn diagram similar to Figure 11 and shade the
region or regions representing the following sets.
(a) A B
Refer to Figure 11. Set A contains all the elements outside of set A, in other
words, the elements in regions 1 and 4. Set B is made up of the elements in regions 3 and 4. The intersection of sets A and B is made up of the elements in
the region common to (1 and 4) and (3 and 4), that is, region 4. Thus, A B is
represented by region 4, which is in color in Figure 12. This region can also be
described as B A.
(b) A B
Again, set A is represented by regions 1 and 4, while B is made up of regions 1
and 2. The union of A and B, the set A B, is made up of the elements belonging to the union of regions 1, 2, and 4, which are in color in Figure 13. An Addison-Wesley product. Copyright © 2004 Pearson Education, Inc.
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71
When the specific elements of sets A and B are known, it is sometimes useful to
show where the various elements are located in the diagram.
EXAMPLE
q
w
B
A
r
B t, v, x.
4
t
s
u
v
x
z
U
Let U q, r, s, t, u, v, w, x, y, z,
A r, s, t, u, v,
y
3
2
1
11
FIGURE 14
Place the elements of these sets in their proper locations on a Venn diagram.
Since A B t, v, elements t and v are placed in region 3 in Figure 14. The
remaining elements of A, that is r, s, and u, go in region 2. The figure shows the
proper placement of all other elements.
To include three sets A, B, and C in a universal set, draw a Venn diagram as in
Figure 15, where again an arbitrary numbering of the regions is shown.
1
B
A
3
2
8
4
5
7
6
C
U
Numbering is arbitrary. The
numbers indicate regions, not
cardinal numbers or elements.
FIGURE 15
E X A M P L E 12 Shade the set A B C in a Venn diagram similar to the
one in Figure 15.
Work first inside the parentheses. As shown in Figure 16, set A is made up of
the regions outside set A, or regions 1, 6, 7, and 8. Set B is made up of regions 1, 2,
5, and 6. The intersection of these sets is given by the overlap of regions 1, 6, 7, 8
and 1, 2, 5, 6, or regions 1 and 6. For the final Venn diagram, find the intersection of
regions 1 and 6 with set C. As seen in Figure 16, set C is made up of regions 4, 5, 6,
and 7. The overlap of regions 1, 6 and 4, 5, 6, 7 is region 6, the region in color in
Figure 16.
1
B
A
3
2
5
8
4
7
C
A
B
3
2
6
4
A B C
FIGURE 16
1
A B is shaded.
(a)
EXAMPLE
13
Is the statement
A B A B
A
2
B
3
4
1
A B is shaded.
(b)
FIGURE 17
true for every choice of sets A and B?
To help decide, use the regions labeled in Figure 11. Set A B is made up of
region 3, so that A B is made up of regions 1, 2, and 4. These regions are in
color in Figure 17(a).
To find a Venn diagram for set A B, first check that A is made up of
regions 1 and 4, while set B includes regions 1 and 2. Finally, A B is made up
of regions 1 and 4, or 1 and 2, that is, regions 1, 2, and 4. These regions are in color
in Figure 17(b).
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The fact that the same regions are in color in both Venn diagrams suggests that
A B A B.
An area in a Venn diagram (perhaps set off in color) may be described using set
operations. When doing this, it is a good idea to translate the region into words,
remembering that intersection translates as “and,” union translates as “or,” and complement translates as “not.” There are often several ways to describe a given region.
De Morgan’s Laws The result obtained in Example 13 is one of De Morgan’s
laws, named after the British logician Augustus De Morgan (1806 – 1871).
De Morgan’s two laws for sets follow.
De Morgan’s Laws
For any sets A and B,
(A B) A B and (A B) A B.
The Venn diagrams in Figure 17 strongly suggest the truth of the first of
De Morgan’s laws. They provide a conjecture, as discussed in Chapter 1. Actual
proofs of De Morgan’s laws would require methods used in more advanced courses
on set theory.
E X A M P L E 14 For each Venn diagram write a symbolic description of the
area in color, using A, B, C, , , , and as necessary.
(a)
A
B
The region in color belongs to all three sets, A
and B and C. Therefore, the region corresponds
to A B C.
C
U
(b)
A
B
C
The region in color is in set B and is not in A
and is not in C. Since it is not in A, it is in A,
and similarly it is in C. The region is, therefore, in B and in A and in C, and corresponds
to B A C.
U
(c) Refer to the figure in part (b) and give two additional ways of describing the
region in color.
The area in color includes all of B except for the regions belonging to either A
or C. This suggests the idea of set difference. The region may be described as
B A C, or equivalently,
B A C.
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