12,774 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

3 votes
3 votes
draw the hasse diag. for above lattice and you can easily figure out the missing key in the puzzle is {1}
3 votes
3 votes

Actually, this qs is about to make COMPLETE Lattice.

If we add {a} then it becomes a complete lattice.

First of all draw the Poset diagram with {a}.And plz don't use the Shortcut for LUB of A and B = A U B and GLB of A and B =A∩B

Just see the Poset diagram and find out the LUB and GLB for every pair of elements.

Let  eg1. subset A= {{1,2},{1,3,5}}  then LUB of {1,2},{1,3,5} = {1,2,3,4,5} ∈ S and GLB of {1,2},{1,3,5} ={1} ∈ S

Let eg2.subset B={{1,2,3},{1,3,5}} then LUB of {1,2,3},{1,3,5} = {1,2,3,4,5} ∈ S and GLB of {1,2,3},{1,3,5} = {1} 

The correct answer is ,(A) {1}

2 votes
2 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.
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
0 votes
0 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 infimum.

We are given a set containment relation.

So,  supremum element is the union of all the subsets and the infinum element is the intersection of all the subsets. Set S is not a complete lattice because although it has a supremum for every subset, some subsets have no infinum.

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.

 

alternative method

 

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hope my answer helps u a lot

edited by
Answer:

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