its greates upper bound
Q)determine whether the poset ({1,2,4,8,16},|)
solution :
i made a hasse diagram of this question but i dont understand how this diagram becomes a lattice.i dont under about least upper bound and lesat lower bound ,,,can 1 is least lower bound and 16 is least upper bound
Let . Let be a nonempty subset. The least upper bound and greatest lower bound of is found by a two step process. First, let and . The set is the set of upper bounds of and the set is the set of lower bounds of . To find the least upper bound and greatest lower bound, let and let . If is nonempty, it must contain exactly one element. Same for . If contains an element, that is the least upper bound. If contains an element, that is the greatest lower bound. In a lattice, both and will be nonempty for any choice of a nonempty subset .
Suppose . Then since and since . Then (so, 2 is the least element of ) since , but 4 does not divide 2, 8 does not divide 2, and 16 does not divide 2. (so, 2 is the greatest element of ) because , but 2 does not divide 1. This means the least upper bound and greatest lower bound of 2 are both 2.
Suppose . Then since , but for any other element of , 16 would not divide it. Hence, 16 is the only upper bound of . Obviously, , so that is the least upper bound. Then and , so 2 is the greatest lower bound.
Suppose . Then the least upper bound will be 8 and the greatest lower bound will be 1.
Etc.