Open In App

GATE | GATE-CS-2004 | Question 73

Like Article
Like
Save
Share
Report

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


Quiz of this Question
Please comment below if you find anything wrong in the above post


Last Updated : 28 Jun, 2021
Like Article
Save Article
Previous
Next
Share your thoughts in the comments
Similar Reads