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}
(B) {1}, {2, 3}
(C) {1}, {1, 3}
(D) {1}, {1, 3}, (1, 2, 3, 4}, {1, 2, 3, 5)


Answer: (A)

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.
Please comment below if you find anything wrong in the above post.

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