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## Homework Statement

Let S=P{2,3,4,6,7,8,14,28,42,98} and let p be the relation on S defined by a p b iff a|b. Then (S,p) is a poset.

(a) Find a subset of S which has no upper bound and no lower bound.

(b) Find the least upper bound for {3,7}

(c) Determine wether or not the subset {2,6,8} of S is totally ordered. Justify your answer.

## Homework Equations

## The Attempt at a Solution

(a) So, I drew a lattice diagram:

http://img514.imageshack.us/img514/8835/72872757.gif [Broken]

I think the set S'={7,3,8} qualifies as a subset of S which has no upper bound and no lower bound. Am I right?

(b) lub{3,7} = 7 ?

(c) I'm not sure how to answer this question. I think that a "total ordering" on a set means a partial order ~ on a set with the additional property that for any [tex]a,b \in S[/tex] we have a~b or b~a. Here in the subset {2,6,8} of S, 6 is devisible by 2, 6|2, also 8|2 but 6 isn't devisible by 8 or vice versa, so the set isn't totally ordered. Is this correct?

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