discrete math

SECTION 10.5

16. Consider the “divides” relation on the following set A. Draw the
Hasse diagram for the relation. (See Overview for drawing tips.)

b. A = {2, 3, 4, 6, 8, 9, 12, 18}

23. Find all greatest, least, maximal, and minimal elements for the
relation in #16b.

42. Use the algorithm given in the text to find a topological sorting
for the relation of exercise #16b that is different from the “less
than or equal to” relation ( . (You only need to write down your
sorting; it is not required to show the steps.)

46. A set S of jobs can be ordered by writing x ( y to mean that either
x = y or x must be done before y, for all x and y is S. Please see Hasse
diagram for this relation on page 601 for a particular set S of jobs:

a. If one person is to perform all the jobs, one after another, find an
order in which the jobs can be done.

b. Suppose enough people are available to perform any number of jobs
simultaneously.

(i) If each job requires one day to perform, what is the least number of
days needed to perform all ten jobs?

(ii) What is the maximum number of jobs that can be performed at the
same time?

47. Suppose the tasks described in Example 10.5.12 require the following
performance times:

Task Time Needed to

Perform Task

1 9 hours

2 7 hours

3 4 hours

4 5 hours

5 7 hours

6 3 hours

7 2 hours

8 4 hours

9 6 hours

a. What is the minimum time required to assemble a car? (Do NOT bother
to turn in the Hasse/PERT diagram. Just indicate what numbers you’ve
added to get the minimum time, and the order in which you added them.)

b. Find a critical path for the assembly process.

48. Section 10.2, #22. Determine whether or not the given binary
relation is reflexive, symmetric, transitive, or none of these. Justify
your answers.

Let SIGMA = {0, 1} and A = SIGMA*. A binary relation G is defined on
SIGMA* as follows:

For all s, t in SIGMA*, s G t iff the number of 0's in s is greater than
the number of 0's in t.

49. Section 10.2, #17. Determine whether or not the given binary
relation is reflexive, symmetric, transitive, or none of these. Justify
your answers.

O is the binary relation defined on Z as follows:

For all m, n in Z, m O n iff m - n is odd.