The inclusion of which of the following sets into
S = {{1, 2}, {1, 2, 3}, {1, 3, 5}, (1, 2, 4), (1, 2, 3, 4, 5}}is necessary and sufficient to make S a complete lattice under the partial order defined by set containment ?
(A)
{1}, {1, 3}
(B)
{1}, {2, 3}
(C)
{1}
(D)
{1}, {1, 3}, (1, 2, 3, 4}, {1, 2, 3, 5)
Answer: (C)
Explanation:
- A partially ordered set L is called a complete lattice if every subset M of L has a least upper bound called as supremum and a greatest lower bound called as infimum.
- We are given a set containment relation.
- So, supremum element is union of all the subset and infimum element is intersection of all the subset.
- Set S is not complete lattice because although it has a supremum for every subset, but some subsets have no infimum.
We take subset {{1,3,5},{1,2,4}}.Intersection of these sets is {1}, which is not present in S.
So we have to add set {1} in S to make it a complete lattice
Thus, option (A) is correct.
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