ICS-241
M. Bauer
Homework for Exam 03
Given the relation on {1,2,3,4} represented by the following zero-one matrix:
1
0

0

0
0 0 1
1 1 0
1 0 1

1 1 1
1) Is it reflexive? _______ If not, what ordered pairs must be added to make it
reflexive?
2) Is it symmetric? ______ If not, what ordered pairs must be added to make it
symmetric?
3) Is it antisymmetric? _______ If not, explain the minimum removals necessary to
make it antisymmetric?
Given the relation on {1,2,3,4} represented by the following directed graph:
1
2
3
4
4) Is it reflexive? _______ If not, what ordered pairs must be added to make it
reflexive?
5) Is it symmetric? ______ If not, what ordered pairs must be added to make it
symmetric?
6) Is it antisymmetric? _______ If not, explain the minimum removals necessary to
make it antisymmetric?
Given the relation on {1,2,3,4} represented by {(1,1), (1,3),(1,4), (2,1), (2,2), (3,1), (3,2),
(3,3) (4,1), (4,4)}
7.) Is it reflexive? _______ If not, what ordered pairs must be added to make it
reflexive?
8) Is it symmetric? ______ If not, what ordered pairs must be added to make it
symmetric?
9) Is it antisymmetric? _______ If not, explain the minimum removals necessary to
make it antisymmetric?
10.)
1
1

0

1
Find M2 for the relation on {1,2,3,4} represented by following zero-one matrix:
1 0 0
1 0 0
1 1 0

0 1 1
11.)
0
1

0

1
Find M2 for the relation on {1,2,3,4} represented by following zero-one matrix:
1 1 0
0 0 1
1 1 1

1 0 1
12.) Given M* for a relation on {1,2,3,4} is represented below, express the transitive
closure as a set of ordered pairs.
1 1 1 1
1 1 0 0


1 0 1 0


1 0 0 1
13.) Given M* for a relation on {1,2,3,4,5} is represented below, express the transitive
closure as a set of ordered pairs.
1
0
0
0
[0
0
1
1
0
0
0
1
1
0
0
0
0
0
1
1
0
0
0
1
1]
14.) Create a zero-one matrix representation of a relation on {1,2,3,4} which is
symmetric.
15.) Create a zero-one matrix representation of a relation on {1,2,3,4} which is
antisymmetric.
16.) Create a set of ordered pairs representation of a relation on {1,2,3,4} which is an
equivalence relation, other than the one given in question 20.
17.) Create a set of ordered pairs representation of a relation on {1,2,3,4} which is a
partial order.
18.) Create a zero-one matrix representation of a relation on {1,2,3,4,5} which is an
equivalence relation, other than the one given in question 20.
19.) Create a zero-one matrix representation of a relation on {1,2,3,4,5} which is a
partial order.
20.) What are the equivalence classes of the following equivalence relation {(1,1), (1,4),
(2,2), (3,3), (3,5), (4,1), (4,4), (5,3), (5,5), (6,6)}?