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.
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Given:
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