Reflexive?
a|a ? Yes.
Symmetric? a|b so b|a? No.
Not equivalence
Option A
Refer: https://gateoverflow.in/62381/go2017-maths-25
Every set of numbers have GCD as Lower Bound, and LCM as upper bound.
Trivial GCD is 1, and trivial LCM is 0.
So, just "upper bound" is asked and not LUB.
Option B seems correct
Existence of relatively prime numbers won't let | be a TOSET.
Option C
As stated above, any infinite subset will have trivial GCD 1 and trivial LCM 0.
But a finite subset like {1, 2, 3} is not a lattice because the pair 2, 3 lacks a join. Source(https://en.wikipedia.org/wiki/Lattice_(order)
So, Option D
A set is called lattice if every finite subset has a least upper bound and greatest lower bound.
The set {1, 2, 3, 12, 18, 36} partially ordered by divisibility is not a lattice. Every pair of elements has an upper bound and a lower bound, but the pair 2, 3 has three upper bounds, namely 12, 18, and 36, none of which is the least.
Source: https://en.wikipedia.org/wiki/Lattice_(order)
Option E
I've spent around 6 hours reading things related to this question, and have come to the following conclusion. The divisibility relation is a POSET (fact) and the divisibility POSET is:-
- If the set is finite, check if it's a lattice or not.
{1, 2, 3, 12, 18, 36} is a not lattice, but {1, 2, 3, 6} is.
- If the set is infinite, 0 can be the Upper Bound and 1 can be the Lower Bound.
So, check if 0 is available in the domain or not.
If you know more about this, please feel free to add a comment.