1
Homework 10 Solutions
University of Pittsburgh
–
Computer Science Department
CS441
–
Discrete Structures for Computer Science
Instructor:
Milos Hauskrecht
Problem 1: Section 9.1
1.
a. {(0, 0), (1, 1), (2, 2), (3, 3)}
b. {(1, 3), (2, 2), (3, 1), (4, 0)}
c. {(1, 0), (2, 0), (2, 1), (3, 0), (3, 1), (3, 2), (4, 0), (4, 1), (4, 2), (4, 3)}
d. {(1, 0), (1, 1), (1, 2), (1, 3), (2, 0), (2, 2), (3, 0), (3, 3), (4, 0)}
e. {(0, 1), (1, 0), (1, 1), (1, 2), (1, 3), (2, 1), (2, 3), (3, 1), (3, 2), (4, 1), (4, 3)}
f. {(1, 2), (2, 1), (2, 2)}
2.
a. (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,2), (2,4), (2,6), (3,3), (3,6), (4,4), (5,5),
(6,6)
b. We draw a line from
a
to
b
whenever
a
divides
b
, using separate sets of points;
an alternate form of this graph would have just one set of points.
c. We put an
×
in the
i
th
row and
j
th
column if and only if
i
divides
j
.
3.
a. Transitive, not reflexive, not symmetric, and not antisymmetric.
b. Transitive, reflexive, symmetric, but not antisymmetric.
c. Not transitive, not reflexive. It is symmetric, but not antisymmetric.
d. Not reflexive, and not symmetric. It is antisymmetric, and it is not transitive.
e. Reflexive and symmetric, trivially antisymmetric and trivially transitive.