Supremum / Infimum of pair of elements on a Hasse Diagram

[Miszka]

Hello! I'm having troubles with one task from Formal Logic and Set Theory that may occur on an exam that I'm taking in just few days. I was given such Hasse Diagram:



I have to find Supremum and Infimum for every pair of points and I did that but I have troubles with some pairs. Mainly:

1) What is infimum for pairs (4, 7), (4, 8) and (6, 8)?
2) What is supremum for pairs (2, 1), (5, 1), (5, 3) and (7, 3)?

I marked sup / inf for all these points as non-existent, but I'm not quite sure about my answer. Can you please help me? I will be very grateful!

 

[pka]

Hello! I'm having troubles with one task from Formal Logic and Set Theory that may occur on an exam that I'm taking in just few days. I was given such Hasse Diagram:

View attachment 21284

I have to find Supremum and Infimum for every pair of points and I did that but I have troubles with some pairs. Mainly:

1) What is infimum for pairs (4, 7), (4, 8) and (6, 8)?
2) What is supremum for pairs (2, 1), (5, 1), (5, 3) and (7, 3)?

I marked sup / inf for all these points as non-existent, but I'm not quite sure about my answer. Can you please help me? I will be very grateful!

Is this question asking for \(\inf\) for each pair or the whole set?

 

[Miszka]

Is this question asking for \(\inf\) for each pair or the whole set?

The question I'm asking is for inf / sup for each pair, that I wrote down. So for the ones below there will be 7 answers consisting of some numbers or saying that inf / sup does not exist. So:

1) What is infimum for pairs (4, 7), (4, 8) and (6, 8)?
2) What is supremum for pairs (2, 1), (5, 1), (5, 3) and (7, 3)?

 

[pka]

The question I'm asking is for inf / sup for each pair, that I wrote down. So for the ones below there will be 7 answers consisting of some numbers or saying that inf / sup does not exist. So:
1) What is infimum for pairs (4, 7), (4, 8) and (6, 8)?
2) What is supremum for pairs (2, 1), (5, 1), (5, 3) and (7, 3)?

\(\bf{\inf}\) is the greatest lower bound while the \(\bf\sup\) is least upper bound.
Thus \(\inf(4,7)=5\) and \(\sup(2,1)=3\)
Please post the rest.

 

[Miszka]

\(\bf{\inf}\) is the greatest lower bound while the \(\bf\sup\) is least upper bound.
Thus \(\inf(4,7)=5\) and \(\sup(2,1)=3\)
Please post the rest.

Thank you. I think that the rest will be:

inf (4, 8) = 3
inf (6, 8) = 7

sup (5, 1) = 7
sup (5, 3) = 4
sup (7, 3) = 8

Is that correct?

 

[pka]

Thank you. I think that the rest will be:
inf (4, 8) = 3
inf (6, 8) = 7

sup (5, 1) = 7
sup (5, 3) = 4
sup (7, 3) = 8

Is that correct?

Definitions do differ. In your textbook(Hesse diagram) is \(4\) an upper bound of \(1~?\)

 

[Miszka]

Definitions do differ. In your textbook(Hesse diagram) is \(4\) an upper bound of \(1~?\)

Unfortunately I don't know, we don't have a book and that topic isn't covered in our lecture notes. We were only finding inf / sup of pairs of elements.

 

[pka]

Unfortunately I don't know, we don't have a book and that topic isn't covered in our lecture notes. We were only finding inf / sup of pairs of elements.

Well then tell me why do you say that \(7=\inf (5,1)~?\) I would have thought the it is \(7\).
Are you given any more information about the Hasse Diagram ? You may be correct, I don't really know what relation the diagram is giving.
It seems that \(4,~6,~7,~\&~8\) are all upper bounds for \((1,5)\) according to the Hasse Diagram. Which of those is least?

 

[Miszka]

Well then tell me why do you say that \(7=\inf (5,1)~?\) I would have thought the it is \(7\).
Are you given any more information about the Hasse Diagram ? You may be correct, I don't really know what relation the diagram is giving.
It seems that \(4,~6,~7,~\&~8\) are all upper bounds for \((1,5)\) according to the Hasse Diagram. Which of those is least?

Unfortunately not. That's the whole task. The problem is: "Find sup and info for every pair of elements" and that's all. And I was saying that sup (5, 1) = 7, because I thought it would work well with what you have said in some posts above and what the definitions said.

 

[pka]

Do you have any idea why in the Hasse Diagram \(7 \prec 6\)?
I really think that whoever wrote this question has left out important information about the relation.
Or maybe somewhere in your notes there is an explanation of how to use the diagram.

 

[Miszka]

Do you have any idea why in the Hasse Diagram \(7 \prec 6\)?
I really think that whoever wrote this question has left out important information about the relation.
Or maybe somewhere in your notes there is an explanation of how to use the diagram.

I contacted my professor regarding that problem. He said that all sups / infs that I asked about (so the same I mentioned in this topic) are non-existent. I think that this is therefore the answer to my problem. Thank you for your support, pka.

 

[pka]

I contacted my professor regarding that problem. He said that all sups / infs that I asked about (so the same I mentioned in this topic) are non-existent. I think that this is therefore the answer to my problem.

Did your instructor actually say that the \(\inf's~\&~\sup's\) did not exist?
Please, ask for an explanation for what the given Hasse Diagram is suppose to be about.
I would greatly like an answer to what the Hasse Diagram means or shows or defines,