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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)
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
The inclusion of which of the following sets intoS = {{1, 2}, {1, 2, 3...
- 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.
Most Upvoted Answer
The inclusion of which of the following sets intoS = {{1, 2}, {1, 2, 3...
To make S a complete lattice under the partial order defined by set containment, we need to include the set {1} into S.
A complete lattice is a partially ordered set in which every subset has a least upper bound (supremum) and a greatest lower bound (infimum). In this case, the partial order is defined by set containment, meaning that one set is considered greater than another if it contains all the elements of the other set.
Let's analyze each option to determine the necessary and sufficient inclusion:
a) {1}
Including the set {1} into S is necessary because it is the smallest element that is not already present in any of the sets in S. Without {1}, S would not have a least element.
b) {1}, {2, 3}
Including {1} and {2, 3} into S is not necessary. {2, 3} is already a subset of {1, 2, 3}, so it does not contribute any additional elements or ordering to S.
c) {1}, {1, 3}
Including {1} and {1, 3} into S is not sufficient to make S a complete lattice. Although {1, 3} is not already present in S, it does not contribute any new ordering relationships between the existing sets.
d) {1}, {1, 3}, (1, 2, 3, 4}, {1, 2, 3, 5)
Including {1}, {1, 3}, (1, 2, 3, 4}, and {1, 2, 3, 5} into S is not necessary. All of these sets are already subsets of {1, 2, 3}, so they do not contribute any additional elements or ordering to S.
Therefore, the correct answer is option 'A' - including the set {1} into S is necessary and sufficient to make S a complete lattice under the partial order defined by set containment.
A complete lattice is a partially ordered set in which every subset has a least upper bound (supremum) and a greatest lower bound (infimum). In this case, the partial order is defined by set containment, meaning that one set is considered greater than another if it contains all the elements of the other set.
Let's analyze each option to determine the necessary and sufficient inclusion:
a) {1}
Including the set {1} into S is necessary because it is the smallest element that is not already present in any of the sets in S. Without {1}, S would not have a least element.
b) {1}, {2, 3}
Including {1} and {2, 3} into S is not necessary. {2, 3} is already a subset of {1, 2, 3}, so it does not contribute any additional elements or ordering to S.
c) {1}, {1, 3}
Including {1} and {1, 3} into S is not sufficient to make S a complete lattice. Although {1, 3} is not already present in S, it does not contribute any new ordering relationships between the existing sets.
d) {1}, {1, 3}, (1, 2, 3, 4}, {1, 2, 3, 5)
Including {1}, {1, 3}, (1, 2, 3, 4}, and {1, 2, 3, 5} into S is not necessary. All of these sets are already subsets of {1, 2, 3}, so they do not contribute any additional elements or ordering to S.
Therefore, the correct answer is option 'A' - including the set {1} into S is necessary and sufficient to make S a complete lattice under the partial order defined by set containment.
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The inclusion of which of the following sets intoS = {{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)Correct answer is option 'A'. Can you explain this answer?
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