Well-ordered principle

[math_newbie85]

Hi everyone. Sorry if this is in the wrong forum (do move this thread if it is, thank you!).

I am going crazy over a principle that seem to be easy and my lecturer is not good at explaining it despite me asking him! The lecture notes are equally insufficient and I have tried to search online before coming here as a last resort.

I have been given two questions:

1. Is set [9, 20) under usual order <= well-ordered?
2. Is set (9, 20] under usual order <= well-ordered?

According to the principle - set S with an order <= is called well-ordered if every non-empty subset of S has a least element.
Axiom - the natural numbers N with the usual order <= is well-ordered.

Now, this are my answer:

1. Yes, as it has the least element 9
2. No, as it does not have the least element

How I approached was that the principle says that it needs to have a least element, in this case, there is 9 inclusive in the first one and 9 exclusive in the second one. I believe I should be trying to see all the possible subsets of the set in order to determine the answer? If yes, how should I approach?

Lastly, should I be concern about any largest element in this principle?

Thank you.

P.S: I am just starting out discrete mathematics, please go easy on me.

 

[pka]

Hi everyone. Sorry if this is in the wrong forum (do move this thread if it is, thank you!).

I am going crazy over a principle that seem to be easy and my lecturer is not good at explaining it despite me asking him! The lecture notes are equally insufficient and I have tried to search online before coming here as a last resort.

I have been given two questions:

1. Is set [9, 20) under usual order <= well-ordered?
2. Is set (9, 20] under usual order <= well-ordered?
According to the principle - set S with an order <= is called well-ordered if every non-empty subset of S has a least element.
Axiom - the natural numbers N with the usual order <= is well-ordered.
Now, this are my answer:
1. Yes, as it has the least element 9
2. No, as it does not have the least element
How I approached was that the principle says that it needs to have a least element, in this case, there is 9 inclusive in the first one and 9 exclusive in the second one. I believe I should be trying to see all the possible subsets of the set in order to determine the answer? If yes, how should I approach?
Lastly, should I be concern about any largest element in this principle?
P.S: I am just starting out discrete mathematics, please go easy on me.

I often tell students to say that the set contains a least element. It means the same thing but emphases membership.
The set \((9,20]\) has a greatest lower bound, \(9\), but does not contain it.
Is the set \((10,20)\subset (9,20)~?\) Does the set \(10,20)\) contain a least term? Do you want to rethink a)?

 

[HallsofIvy]

You may be misunderstanding the notation.
[a, b] is the set \(\displaystyle \{x| a\le x\le b\}\).
Its minimum is a and its maximum is b.

(a, b) is the set \(\displaystyle \{x| a< x< b\}\).
It has NO minimum or maximum.

[a, b) is the set \(\displaystyle \{x| a\le x< b\}\).
Its minimum is a but it has no maximum.

(a, b] is the set \(\displaystyle \{x| a< x \le b\}\).
It has no minimum but its maximum is b.

 

[math_newbie85]

I often tell students to say that the set contains a least element. It means the same thing but emphases membership.
The set \((9,20]\) has a greatest lower bound, \(9\), but does not contain it.
Is the set \((10,20)\subset (9,20)~?\) Does the set \(10,20)\) contain a least term? Do you want to rethink a)?

Thanks for your respond. Hope you don't mind me working out on your question (hope this is the right approach too).

For set \((10,20)\subset (9,20)~\), the symbol means a proper subset which is \(A \subset B\); A is a subset of B, but A is not equal to B.

\((10,20)\) = \( \{ 11, 12, 13, 14, 15, 16, 17, 18, 19 \} \)
\((9,20)\) = \( \{ 10, 11, 12, 13, 14, 15, 16, 17, 18, 19\} \)

Therefore, \((10,20)\subset (9,20)~\).

The set \((10,20)\) do not contain a least element or maximum element (I am basing this on HallsofIvy reply).

P.S: I hope I this working is correct. Please enlighten me!

You may be misunderstanding the notation.
[a, b] is the set \(\displaystyle \{x| a\le x\le b\}\).
Its minimum is a and its maximum is b.

(a, b) is the set \(\displaystyle \{x| a< x< b\}\).
It has NO minimum or maximum.

[a, b) is the set \(\displaystyle \{x| a\le x< b\}\).
Its minimum is a but it has no maximum.

(a, b] is the set \(\displaystyle \{x| a< x \le b\}\).
It has no minimum but its maximum is b.

Thank you. This means that:

[9, 20) - Its minimum is 9 but HAS NO maximum
(9, 20] - It has no minimum but HAS A maximum 20

Getting more confused at well-ordered principle. Does the greatest element play a role in the principle as the principle only mention the least element. What will be the best way to describe the reason why it is well-ordered?

My initial post answers was:

[9, 20) - It has a least element 9
(9,20] - It does not have the least element

In this case, what will be the best way to describe the two sets? Do I have to mention both the least element and maximum element? What I am thinking of that best describe these two cases are:

[9, 20) - It has a least element 9

(9,20] - It has the greatest element 20

This answer is following according to what you mentioned and by excluding the side that is exclusive.

I feel that something is still missing in my brain on this!

 

[Steven G]

Thanks for your respond. Hope you don't mind me working out on your question (hope this is the right approach too).

For set \((10,20)\subset (9,20)~\), the symbol means a proper subset which is \(A \subset B\); A is a subset of B, but A is not equal to B.

\((10,20)\) = \( \{ 11, 12, 13, 14, 15, 16, 17, 18, 19 \} \)
\((9,20)\) = \( \{ 10, 11, 12, 13, 14, 15, 16, 17, 18, 19\} \)

No no no! (9,20) is NOT {10,11,12,13,14,15,16,17,18,19} NO! What about sqrt(101) and 29/4 and 11.124?
(9,20)= {x | 9<x<20} which is read as the set of ALL x such that x is between 9 and 20 and not including either end point.

 

[math_newbie85]

No no no! (9,20) is NOT {10,11,12,13,14,15,16,17,18,19} NO! What about sqrt(101) and 29/4 and 11.124?
(9,20)= {x | 9<x<20} which is read as the set of ALL x such that x is between 9 and 20 and not including either end point.

Thanks Jomo for the reply.

{x | 9<x<20} is a set notation whereby any positive integer can be in the range of 10 to 19. Hence it can be any set of x, am I right? I believe it is impossible to list all the sets since it can literally be any integers between the range?

From the responses so far, I understand that this is a range of x integers whereby ( is exclusive and [ is inclusive. The set can be of any integers between this range. Then the question boils down again to what is the best way to describe the two interval notation? And also whether the greatest element can be use to describe the sets since the principle stated least element? Also does usual order provide a certain key hint to describe?

I know this isn't right but perhaps if I post the options I am given to the question it will let you guys see why I may be getting confused:

Is set [9,20) under usual order <= well-ordered? Why?
1) Yes, as it has the least element 9 and greatest element 20
2) No, as it does not have the greatest element
3) Yes, as it has the least element 9 (My answer selected is this, reason - 9 is the minimum since it is inclusive)
4) Yes, as it has the least element 9 and greatest element 19

Is the set (9,20] under usual order <= well-ordered? Why?
1) No, as it does not have both the least and greatest elements
2) No, as it does not have the least element
3) Yes, as it has the greatest element 20 (My answer selected is this, reason - 20 is maximum since it is inclusive)
4) Yes, as it has the least element 9

This is why I am wondering if the greatest element has to do with the principle since the principle stated least element so shouldn't (9,20] answer suppose to be (2)?

 

[JeffM]

This is not an area of mathematics on which I usually comment. But why do you say that the set contains only integers? Nothing that you have said so far about the problem seems to support that restriction. Integers are discrete. Real numbers are not.

 

[pka]

This is not an area of mathematics on which I usually comment. But why do you say that the set contains only integers? Nothing that you have said so far about the problem seems to support that restriction. Integers are discrete. Real numbers are not.

I suspect that we are speaking at cross purposes here. It appears that this question comes from a discrete mathematics course. The operative word here is discrete as opposed to continuous. The set of real numbers is not well ordered. Any set of integers bounded below is well order. That is the "stuff" of discrete mathematics. Therefore. I think this question is about discrete subsets of real real numbers.
I think the student poster is not clear on this difference. If I am wrong, someone please correct!

 

[math_newbie85]

I suspect that we are speaking at cross purposes here. It appears that this question comes from a discrete mathematics course. The operative word here is discrete as opposed to continuous. The set of real numbers is not well ordered. Any set of integers bounded below is well order. That is the "stuff" of discrete mathematics. Therefore. I think this question is about discrete subsets of real real numbers.
I think the student poster is not clear on this difference. If I am wrong, someone please correct!

Hi there. Yes, apology if I wasn't clear on this earlier on. This is indeed discrete mathematics! Will like to apology if this is not the right forum for such question.

 

[Steven G]

We really do need a discrete math sections. Hint hint.

 

[Steven G]

Thanks Jomo for the reply.

{x | 9<x<20} is a set notation whereby any positive integer can be in the range of 10 to 19. Hence it can be any set of x, am I right? I believe it is impossible to list all the sets since it can literally be any integers between the range?

From the responses so far, I understand that this is a range of x integers whereby ( is exclusive and [ is inclusive. The set can be of any integers between this range. Then the question boils down again to what is the best way to describe the two interval notation? And also whether the greatest element can be use to describe the sets since the principle stated least element? Also does usual order provide a certain key hint to describe?

I take it that you do not know what an integer is. The integers = {..., -3, -2, -1, 0, 1, 2, 3,...}. There are many numbers between any two consecutive integers AND these numbers are NOT integers. There are no integers between 7 and 8!! There are real numbers between 7 and 8. What are real numbers. Any number (except imaginary numbers, ie complex numbers) is a real number. 7, pi, sqrt(11), 6.456 are all real numbers.

{x | 9<x<20} is a set notation of any positive integer in the range from 10 to 19 IS WRONG. {x | 9<x<20} is not the same as {10, 11, 12, 13,14, 15, 16, 17, 18, 19}. Stop saying integers when you should not be saying integers.

 

[math_newbie85]

I take it that you do not know what an integer is. The integers = {..., -3, -2, -1, 0, 1, 2, 3,...}. There are many numbers between any two consecutive integers AND these numbers are NOT integers. There are no integers between 7 and 8!! There are real numbers between 7 and 8. What are real numbers. Any number (except imaginary numbers, ie complex numbers) is a real number. 7, pi, sqrt(11), 6.456 are all real numbers.

{x | 9<x<20} is a set notation of any positive integer in the range from 10 to 19 IS WRONG. {x | 9<x<20} is not the same as {10, 11, 12, 13,14, 15, 16, 17, 18, 19}. Stop saying integers when you should not be saying integers.

Thanks Jomo. Sorry if this is confusing you and any others (math isn't my strong subject since early school days). I believe what confused everyone here is the usage of integers and real number; which I understand from your point that these two are different. I learnt it recently but am too used to calling any numbers an integers due to habit.

It will have been clearer if the questions indicate whether the value of x is an integer, Z OR real number, R. Given everyone answers, I have come to a conclusion that the answers for my question will be (3) Yes, as it has the least element 9 and (2) No, as it does not have the least element as everyone has pointed out the inclusive and exclusive of the range and the reasoning of greatest element do not come into play here.