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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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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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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. 

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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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