Question

Answer the following questions concerning the poset ({3, 5, 9, 15, 24, 45}, D (the divisibility relation on the set) 1. Find the maximal elements 3 5 9 15 24 45 2. Find the minimal elements 3 5 9 15 24 45 3. Find the minimum, if it exists 3 5 9 15 24 45 none 4. Find the maximum, if it exists 03 05 09 15 24 45 none 5. Find all upper bounds of {3,5} 3 5 9 15 24 45 none 6. Find the least upper bound of {3,5} if it exists 3 5 9 15 24 45 none 7. Find all lower bounds of {15,45} 3 5 9 15 24 45 none 8. Find the greatest lower bound of {15,45} if it exists. 03 5 9 15 24 45 none

Answer 1

1. The maximal elements of the poset are: 3, 5, 9, 15, 24, 45.
2. The minimal elements of the poset are: 3, 5, 9, 15, 24, 45.
3. The minimum does not exist since there is no element in the set that is less than or equal to all other elements in the set.
4. The maximum does not exist since there is no element in the set that is greater than or equal to all other elements in the set.
5. All upper bounds of {3,5} are: 9, 15, 24, 45.
6. The least upper bound of {3,5} is 15.
7. All lower bounds of {15,45} are: 3, 5, 9.
8. The greatest lower bound of {15,45} is 15.