I've been thinking about the following hypothetical problem given in a discrete math textbook (with no explanations), but don't know if I'm going in the right direction.
Given the poset:
⟨{{A}, {B}, {D}, {A, B}, {A, D}, {C, D}, {A, B, D}, {B, C, D}}, ⊆⟩
is it possible to find examples of:
(a) Two elements in the poset that have no lower bound?
- Here I think that since the elements {A}, {B}, and {D} are sets with a single element, with: {A} and {B}, {B} and {D}, or {A} and {D} as examples of pairs which satisfy this because since they are not necessarily related to one another, they cannot be subsets of one another. However they can be a subset of themselves, as well as the empty set.
(b) Two elements in the poset that have lower bounds, but no greatest lower bound?
- In the same line of thinking as the previous part, it would have to be both of the elements {A, D} or {C, D}?