Lattice : A partially ordered set in which every pair of elements has both a least upper bound and a greatest lower bound is called as lattice.
Upper Bound: An element which is greater than or equal to all the element in a subset is called upper bound of that subset.
Lower Bound: An element which is less than or equal to all the element in a subset is called lower bound of that subset.
Least Upper Bound : An element x is called the least upper bound of a pair (a,b) If x is an upper bound that is less than every other upper bound of (a, b).
Greatest Lower Bound : An element x is called the greatest lower bound of a pair (a, b). If x is a lower bound that is greater than every lower bound of (a,b).
In A, element 2 and 3 doesn't have least upper bound. Upper bound of 2 and 3 are 4, 5 and 6 But none of them is least upper bound. 6 is greater than 4 and 5. But 4 and 5 are not comparable. So we don't know which is less.
Similarly in D, 2 and 3 doesn't have Least upper bound. Upper bound of 2 and 3 are 7, 6 and 8.
4 and 5 are not upper bound of 2 and 3. Because 2 and 5 are not comparable we don't know which is greater. Similarly between 3 and 4 we don't know which one is greater. Among 6, 7 and 8, 8 is greater than 6 and 7. But 6 and 7 are not comparable. So again we don't have least upper bound.
For B and C, we have greatest lower bound and least upper bound for every pair of element so they are lattices.
More ever Least upper bound and greatest lower bound for a subset is unique if exist.
