12,768 views
42 votes
42 votes

The inclusion of which of the following sets into

$S = \left\{ \left\{1, 2\right\}, \left\{1, 2, 3\right\}, \left\{1, 3, 5\right\}, \left\{1, 2, 4\right\}, \left\{1, 2, 3, 4, 5\right\} \right\} $

is necessary and sufficient to make $S$ a complete lattice under the partial order defined by set containment?

  1. $\{1\}$
  2. $\{1\}, \{2, 3\}$
  3. $\{1\}, \{1, 3\}$
  4. $\{1\}, \{1, 3\}, \{1, 2, 3, 4\}, \{1, 2, 3, 5\}$

8 Answers

Best answer
34 votes
34 votes

Answer: A

A lattice is complete if every subset of partial order set has a supremum and infimum element.

For example, here we are given a partial order set S. Now it will be a complete lattice if whatever be the subset we choose, it has a supremum and infimum element. Here relation given is set containment, so supremum element will be just union of all sets in the subset we choose. Similarly, infimum element will be just intersection of all the sets in the subset we choose.

Now as we can see, $S$ now is not complete lattice, because although it has a supremum for every subset we choose, but some subsets have no infimum. For example: if we take subset $\{1,3,5\}$ and $\{1,2,4\}$ then intersection of sets in this is $\{1\},$ which is not present in $S.$ So clearly, if we add set $\{1\}$ in $S,$ we will solve the problem. So, adding $\{1\}$ is necessary and sufficient condition for $S$ to be complete lattice. Thus, option (A) is correct.

75 votes
75 votes

See the below diagram only {1} is enough to be Lattice. Hence Option A is Ans.

12 votes
12 votes
given set contains two infimum (1,2) & (1,3,5) so given set is not a lattice.
adding (1) to the given set results lattice..
12 votes
12 votes
  • 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
Answer:

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