KINDLY HELP Answer the following questions for the poset 2 4

KINDLY HELP

Answer the following questions for the poset ({2, 4, 6, 9, 12, 18, 27, 36, 48, 60, 72}, 1). What are the maximal elements? What are the minimal elements? Is there a greatest element? Is there a least element? What are the upper bounds of {2, 9}? What is the least upper bound of {2, 9}, if it exists? What are the lower bound of {2, 9}, if it exists? What are the lower bounds of {60, 72}? What is the greatest lower bound of {60, 72}, if it exists?

Solution

a) Maximal elements: 27, 48, 60, 72 do not divide anything higher in the set, so theyare maximal.

b) Minimal elements: 2, 9 are not divisible by other elements in the set, soare minimal.

c) Greatest element:There is no element in the set which is divisible by everyother element in the set.

d) Nothing divides every element of the set, for 2 and 9 are coprime.

f) 18 is divisible by both 2 and 9, and is the least such element, both theoretically, and in the set.