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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}?
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i) (b) doesn't make sense; is it meant to read "that have a lower bound but no greatest lower bound"? ii) You seem to have misunderstood the questions. "Two elements that have no (greatest) lower bound" refers to a common (greatest) lower bound of the two elements, so you need to exhibit pairs of elements. – joriki Nov 28 '12 at 16:01
I meant to say elements that have lower bounds, but no greatest lower bound - see the edit. – verovdp Nov 28 '12 at 16:06
Is this a verbatim quote from the textbook? Does it really say "lower bounds" in the plural? The question would make more sense if it said "a lower bound", as I'd suggested. – joriki Nov 28 '12 at 16:10
The question specifically stands "lower bounds" in the plural – verovdp Nov 28 '12 at 16:35
That's strange. Every element by itself has a lower bound, namely itself, so it doesn't really make sense that way; it would only make sense if the two elements had one common lower bound. – joriki Nov 28 '12 at 16:38
up vote 1 down vote accepted

(a) You’re correct: any two of $\{A\},\{B\}$, and $\{D\}$ are an example of two elements of the poset with no lower bound. The pair $\big\{\{A,B\},\{C,D\}\big\}$ also works, as does the pair $\big\{\{A\},\{B,C,D\}\big\}$.

(b) The elements $\{A,D\}$ and $\{C,D\}$ do have a greatest lower bound, namely, $\{D\}$. However, the pair $\big\{\{A,B,D\},\{B,C,D\}\big\}$ does satisfy the specified conditions: it has $\{B\}$ and $\{D\}$ as lower bounds, and neither of these is a subset of the other, so it has no greatest lower bound.

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