166
CHAPTER 3 Relations
Exercises
1. For the people in the family tree (see Figure 3.2), build tables for the following relations:
(a) IsAncestorOf
(b) IsDescendentOf
(c) IsSiblingOf
(d) IsCousinOf
2. Let M = {1, 2 ... , 10}. Define a relation R on elements x, y E M such that (x, y) E
R if and only if there is a positive integer k such that x = ky. Find the elements of R.
3. Find the elements in each of the following relations defined on R:
(a) (x, y) E R if and only if x + 1
(c) (x, y, z) E R if and only ifx
2 + y = z
4. List the 16 elements of the relation Between as defined in Example 4.
5. The table below gives the names of airlines and several cities that each flies to from
Chicago. The table also gives the number of miles for each flight. List all the triples
(X, Y, Z) of the ternary relation defined by those triples for which airline X flies Y
miles to city Z.
TWA
Pan Am
Piedmont
Topeka
603
Bombay
7809 Peoria
170
Kansas City
510 Seattle
2052 Albany
816
Phoenix
1742 Anaheim 2025
Atlanta
717
6. Let U = (0, 11.
(a) Let SubsetOf = {(X, Y) : X, Y C U and X C Y}. List all ordered pairs in SubsetOf.
(b) Let StrictSubsetOf= {(X, Y) : X, Y C U and X C Y 1. List all its ordered pairs in
StrictSubsetOf.
(c) {(X, Y, Z) : X, Y, Z C U and X n Y = ZJ is a ternary relation. List all ordered
triples in this relation.
The relations SubsetOf and StrictSubsetOf can be defined on any set of sets. We will
use these relations for other universal sets later in the text.
7. Using the family tree shown in Figure 3.2, list the elements in each of the following
relations, and give these relations meaningful names.
(a) IsMarriedTo -1
(b) IsMarriedTo o IsMarriedTo
(c) IsParentOf o IsParentOf-1
(d) =Family where Family denotes the set of people appearing in the family tree
(e) IsMarriedTo n IsMarriedTo
1
(f) IsParentOf n IsParentOf-1
8. Describe the relations resulting from the inverse or composition operations. Describe
the resulting relations in words.
CHAPTER 3 Relations
Exercises
1. For the people in the family tree (see Figure 3.2), build tables for the following relations:
(a) IsAncestorOf
(b) IsDescendentOf
(c) IsSiblingOf
(d) IsCousinOf
2. Let M = {1, 2 ... , 10}. Define a relation R on elements x, y E M such that (x, y) E
R if and only if there is a positive integer k such that x = ky. Find the elements of R.
3. Find the elements in each of the following relations defined on R:
(a) (x, y) E R if and only if x + 1
(c) (x, y, z) E R if and only ifx
2 + y = z
4. List the 16 elements of the relation Between as defined in Example 4.
5. The table below gives the names of airlines and several cities that each flies to from
Chicago. The table also gives the number of miles for each flight. List all the triples
(X, Y, Z) of the ternary relation defined by those triples for which airline X flies Y
miles to city Z.
TWA
Pan Am
Piedmont
Topeka
603
Bombay
7809 Peoria
170
Kansas City
510 Seattle
2052 Albany
816
Phoenix
1742 Anaheim 2025
Atlanta
717
6. Let U = (0, 11.
(a) Let SubsetOf = {(X, Y) : X, Y C U and X C Y}. List all ordered pairs in SubsetOf.
(b) Let StrictSubsetOf= {(X, Y) : X, Y C U and X C Y 1. List all its ordered pairs in
StrictSubsetOf.
(c) {(X, Y, Z) : X, Y, Z C U and X n Y = ZJ is a ternary relation. List all ordered
triples in this relation.
The relations SubsetOf and StrictSubsetOf can be defined on any set of sets. We will
use these relations for other universal sets later in the text.
7. Using the family tree shown in Figure 3.2, list the elements in each of the following
relations, and give these relations meaningful names.
(a) IsMarriedTo -1
(b) IsMarriedTo o IsMarriedTo
(c) IsParentOf o IsParentOf-1
(d) =Family where Family denotes the set of people appearing in the family tree
(e) IsMarriedTo n IsMarriedTo
1
(f) IsParentOf n IsParentOf-1
8. Describe the relations resulting from the inverse or composition operations. Describe
the resulting relations in words.