**Unit 22 Exercises**

**1. **Which of the following are posets?

- (
**Z**, = ) - (
**Z**, ) - ( A collection of sets, ).

**2. **Draw the Hasse diagram for the divisibility relation
on the following sets

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

b) {1, 2, 5, 8, 16, 32}

**3.** Answer the following questions concerning the poset
({{1}, {2}, {3}, {4}, {1, 3}, {1, 4}, {2, 4}, {3, 4}, {1, 2, 4}, {2, 3, 4}},
).

- Find the maximal elements.
- Find the minimal elements.
- Is there a greatest element?
- Is there a least element?
- Find all upper bounds of {{2}, {4}}.
- Find the least upper bound of {{2}, {4}}, if it exists.
- Find all lower bounds of {{1, 2, 4}, {2, 3, 4}}
- Find the greatest lower bound of {{1, 2, 4}, {2, 3, 4}}, if it exists.