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Discrete Math and Divides in Relation

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.

Solution Preview

Solution.
<br>
<br>(16) The Hasse Diagram is as follows.There are four lines added which are from 2 to 6, 3 to 9, 4 to 12, and 6 to 18 respectively.
<br>
<br> 2------->4-------->8
<br>
<br> 3------->6-------->12
<br>
<br> 9-------->18
<br>
<br>
<br>(23) Solution.
<br> The greatest element is 18 since there is no element which bigger than 18 and can be divided by 18. So 2 should be the least one.
<br>The maximal elements are 8, ...

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