# Hasse diagram, ordered sets, sorting

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 in S. The following is a Hasse diagram for this relation for a particular set S of jobs:

(see attached)

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 the jobs?

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

#### Solution Preview

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

({1,10},{6},{5,9,7},{3,4},{2,8})

Any of the ordered sequence is OK eg. 10,1,6,5,7,9,3,4,2,8 as long ...