Answer these questions for the poset ({3, 5, 9, 15, 24, 45}). a) Find the maximal elements. b) Find the minimal elements. c) Is there a greatest element? d) Is there a least element? e) Find all upper bounds of { 5, 9}. f ) Find the least upper bound of { 5, 9}, if it exists. g) Find all lower bounds of {15, 24}. h) Find the greatest lower bound of {15, 24}, if it exists.

Given:

 $\begin{aligned} &(\{3,5,9,15,24,45\}, \mid) \\ &S=\{3,5,9,15,24,45\} \\ &R=\{(a, b) \mid a \text { divides } b\} \end{aligned}$

Let us first determine the Hasse diagram

(a) The maximal elements are all values in the Hasse diagram that do not have any elements above it.

maximal elements = 24,45

(b) The minimal elements are all values in the Hasse diagram that do not have any elements below it. 

minimal elements = 3,5

(c) The greatest element only exists is there is exactly one maximal element and is then also equal to that maximal element.

greatest element = Does not exist

(d) The least element only exists is there is exactly one minimal element and is then also equal to that minimal element.

least element = Does not exist

(e) The upper bounds of a set are all elements that have a downward path to all elements in the set.

upper bounds =15,45

(f) The least upper bound of a set is the upper bound that is less than all other upper bounds.

least upper bound =15

(g) The lower bounds of a set are all elements that have an upward path to all elements in the set. 

lower bound = 3,5,15

(h) The greatest lower bound is the lower bound that is greater than all other upper bounds. 

greatest lower bound = 15