3.3 Exercises
1. For the people in the family tree (see Figure 32), build tables for the
following relations:
(a)
IsAncestorOf
(b)
IsDescendentOf
(c)
IsSiblingOf
(d)
IsCousinOf
1.
= John
Mary
Maude=Harold
Peter=Elaine
George
Elizabeth
1. (a)
IsAncestorOf
Elaine
George
Mary
George
John
George
Peter
George
Harold
Elizabeth
Maude
Elizabeth
Mary
Elizabeth
John
Elizabeth
Mary
Elaine
John
Elaine
Mary
Maude
John
Maude
1. (b)
IsDescendentOf
George
Mary
George
John
George
Peter
George
Elaine
Maude
Mary
Maude
John
Elizabeth
Maude
Elizabeth
Harold
Elizabeth
Mary
Elizabeth
John
Elaine
Mary
Elaine
John
1. (c)
IsSiblingOf
Elaine
Maude
Maude
Elaine
1. (d)
IsCousinOf
George
Elizabeth
Elizabeth
George
3. Find the elements in each of the following relations defined on
R
:
(a) (
x, y
)
∈
R
if and only if
x
+ 1
< y
(b) (
x, y
)
∈
R
if and only if
y <
0 or 2
x
≤
3
(c) (
x, y, z
)
∈
R
if and only if
x
2
+
y
=
z
3. (a)
3. (b)
3. (c)
2
0
2
2
0
2
0
5
10
5. The table 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
5.
Airlines
Mileage
City
TWA
603
Topeka
TWA
510
Kansas City
TWA
1742
Phoenix
Pan Am
7809
Bombay
Pan Am
2052
Seattle
Pan Am
2025
Anaheim
Piedmont
170
Peoria
Piedmont
816
Albany
Piedmont
717
Atlanta
7. Using the family tree shown in Figure 32, list the elements in each of the
following relations, and give these relations meaningful names.
(a)
IsMarriedTo

1
(b)
IsMarriedTo
◦
IsMarriedTo
(c)
IsParentOf
◦
IsParentOf

1
(d) =
Family
where
Family
denotes the set of people appearing in the
family tree
(e)
IsMarriedTo
∩
IsMarriedTo

1
(f)
IsParentOf
∩
IsParentOf

1
7. (a)
IsMarriedTo

1
John
Mary
Mary
John
Peter
Elaine
Elaine
Peter
Maude
Harold
Harold
Maude
7. (b)
IsMarriedTo
◦
IsMarriedTo
Peter
Peter
Elaine
Elaine
Maude
Maude
Harold
Harold
Mary
Mary
John
John
7. (c)
IsParentOf
◦
IsParentOf

1
Elaine
Elaine
Elaine
Maude
Maude
Elaine
Maude
Maude
George
George
Elizabeth
Elizabeth
7. (d)
=
Family
Peter
Peter
Elaine
Elaine
Mary
Mary
John
John
Maude
Maude
Harold
Harold
Elizabeth
Elizabeth
George
George
7. (e)
IsMarriedTo
∩
IsMarriedTo

1
John
Mary
Mary
John
Peter
Elaine
Elaine
Peter
Maude
Harold
Harold
Maude
7. (f)
∅
9. Prove Theorem 2(c).
9.
For any
x, y
∈
X
such that (
x, y
)
∈
S

1
, it follows that (
y, x
)
∈
S.
Since
S
⊆
R,
(
y, x
)
∈
R.
Now, (
x, y
)
∈
R

1
.
11. Let
A
=
{
1
,
2
,
3
, . . .,
10
}
.
Let
R
=
{
(1, 2), (1, 4), (1, 6), (1, 8), (1,
10), (3, 5), (3, 7), (4, 6), (6, 8), (7, 10)
}
be a relation on
A.
Let
S
=
{
(2
,
4)
,
(3
,
6)
,
(5
,
7)
,
(7
,
9)
,
(8
,
10)
,
(8
,
9)
,
(8
,
8)
,
(9
,
9)
,
(3
,
8)
,
(4
,
9)
}
be a sec
ond relation on
A.
Find:
(a)
R
◦
S
(b)
S
◦
R.
11. (a)
{
(2
,
6)
,
(3
,
8)
,
(5
,
10)
}
11. (b)
{
(1
,
4)
,
(1
,
9)
,
(1
,
10)
,
(1
,
8)
,
(3
,
7)
,
(3
,
9)
,
(6
,
10)
,
(6
,
9)
,
(6
,
8)
}
13. Let
R, S
and
T
be binary relations on a set
X
.
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 Summer '10
 Halpern