# Upper/Lower Bounds

## 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.

## 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 $$a,b \in S$$ 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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## Answers and Replies

tiny-tim
Science Advisor
Homework Helper
Hi roam! 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.

(a) … 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 $$a,b \in S$$ 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?

(a) yes (b) nooo (c) yes Hi roam! (a) yes (b) nooo (c) yes Hi Tiny tim!

I don't understand why my answer to part (b) is not correct, could you explain please? 3x7=21 which is not even in the bigger set, and the only (and hence smallest) upperbound is 7. Also for part (c), is my explanation correct/sufficient (because the question says you must justify the answer)?

Office_Shredder
Staff Emeritus
Science Advisor
Gold Member
How can 7 be the least upper bound if 3 isn't less than 7? You say 21 isn't in S, and that's fine... what numbers in S could be the least upper bound?

tiny-tim
Science Advisor
Homework Helper
Hi roam! I don't understand why my answer to part (b) is not correct, could you explain please? 3x7=21 which is not even in the bigger set, and the only (and hence smallest) upperbound is 7. Office_Shredder is right … 3 does not divide 7 (ie, is not "less than" in the ordering), so how can 7 be the lub?
Also for part (c), is my explanation correct/sufficient (because the question says you must justify the answer)?

Yes … you've specifically said that "6 isn't divisible by 8 or vice versa", which itself shows that there is no total ordering … that's fine! The set has no upper/lower bound since there is no order on the set, 3 & 7 aren't devisible. So is it just the empty set i.e. lub{3,7}= {$$\emptyset$$}? Tiny tim, there's another question that I'm confused about: "Find all maximal and minimal elements of S."
It says "elements", plural. But there's only one max and one min:
maxP{2,3,4,6,7,8,14,28,42,98}=98
minP{2,3,4,6,7,8,14,28,42,98}=2

tiny-tim
Science Advisor
Homework Helper
Hi roam! (oh, and it's "divisible"! )
The set has no upper/lower bound since there is no order on the set, 3 & 7 aren't devisible. So is it just the empty set i.e. lub{3,7}= {$$\emptyset$$}? Forget division … just look at that very good picture you drew … go up from 3 and 7, and where do you get? (btw, lub has to be an element

if there's no lub, you just say so, you don't say it's the empty set. )

Tiny tim, there's another question that I'm confused about: "Find all maximal and minimal elements of S."
It says "elements", plural. But there's only one max and one min:
maxP{2,3,4,6,7,8,14,28,42,98}=98
minP{2,3,4,6,7,8,14,28,42,98}=2

I haven't come across this terminology before , but I assume maximal means any element that doesn't have anything "higher" …

again, you can just read this off the picture. Thanks alot Tiny tim, I get it! Forget division … just look at that very good picture you drew … go up from 3 and 7, and where do you get?

lub=42

tiny-tim
Science Advisor
Homework Helper Woohoo! 