Need to know how to do directed sets

Cratylus

Cratylus

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1. R with <
2. R with <=
3. Z with <

Show which of the above is a directed set. For the ones that aren’t explain why it isnt

Amy help is appreciated.
 
Cratylus said:
1. R with <
2. R with <=
3. Z with <

Show which of the above is a directed set. For the ones that aren’t explain why it isn't

Amy help is appreciated.
Click to expand...
Please show what you have tried, so we can see where you need help. State the definition you are using, and tell us how you think it does or doesn't apply to one of these cases. What is it that is confusing you? (I'm going by this definition.)

I don't think any of us are named Amy, but I imagine you'll appreciate help from any of us. ;)
 
That is a typo .

Any help would be appreciated.i am using that definition too.

According to my text, A first course in topology by Conover (3) is a directed set .

For. Transitivity, let x,y[imath]\in[/imath] Z ,if x<y and y<z then xRy and yRz implies xRz so Z is transitive.

reflexive is obvious

So I have to find z>=x and z>=y I think this is UB property. Z has the UB property
so is a directEd set

.
 
Cratylus said:
According to my text, A first course in topology by Conover (3) is a directed set .
For. Transitivity, let x,y[imath]\in[/imath] Z ,if x<y and y<z then xRy and yRz implies xRz so Z is transitive.
reflexive is obvious
So I have to find z>=x and z>=y I think this is UB property. Z has the UB property
so is a directEd set.
Click to expand...
Why is it that you always refuse all requests to supply exact definitions?
There is no universal agreement on what a directed set is.
The MathWorld Website requires that such ordering be both transitive & reflexive.
Well that rules out [imath]<[/imath] as a possible direct ordering.
Moreover, the classily trained Charles Pinter(Univ of Paris) states that [imath]<[/imath] is not an order relation.
Thus, unless you give the exact definition in use there little we can help.

[imath][/imath][imath][/imath][imath][/imath][imath][/imath]
 
Dr.Peterson said:
Can you show an image of what it says?


You still need to show it; please do so. Which relation are you referring to when you say this?

I think you may be missing something very simple.
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(3) . What am I missing . I worked with Pinter’s book a book of set theory. Made it up to chapter 7, before he lost me.

to show (3) is reflexive,let x[imath]\in[/imath] Z,then xRx implies x=xC579DBA9-4B56-482C-A0B2-FC88F213F1D1.jpeg
Dr.Peterson said:
Can you show an image of what it says?


You still need to show it; please do so. Which relation are you referring to when you say this?

I think you may be missing something very simple.
Click to expand...
 
pka said:
Why is it that you always refuse all requests to supply exact definitions?
There is no universal agreement on what a directed set is.
The MathWorld Website requires that such ordering be both transitive & reflexive.
Well that rules out [imath]<[/imath] as a possible direct ordering.
Moreover, the classily trained Charles Pinter(Univ of Paris) states that [imath]<[/imath] is not an order relation.
Thus, unless you give the exact definition in use there little we can help.

[imath][/imath][imath][/imath][imath][/imath][imath][/imath]
Click to expand...
I am using MathWorld Website defs as mentioned by Dr Peterson
 
Cratylus said:
3. Z with <
Click to expand...
Cratylus said:
to show (3) is reflexive,let x∈Z,then xRx implies x=x
Click to expand...
Really? You quoted this definition of reflexive:
1663369417391.png
That's not what you are saying! Your version doesn't make sense.

Now, are you saying that < is reflexive? By the correct definition, you are saying that any number is less than itself. Not true.

Cratylus said:
According to my text, A first course in topology by Conover (3) is a directed set .
Click to expand...
What I asked you to show was an image of how the author says this. I want to see how the book says that #3 is a directed set.

I don't even see #3 as one of the exercises you've shown; the closest is
1663369695649.png
But that's a different set, and, more important, a different relation.

My best guess is that you are (a) not reading carefully, and/or (b) not copying carefully.
 
Do not confuse symbols [imath]<[/imath] is used for less than and is not reflexive.
While [imath] \bf\large\prec[/imath] means precedes and may be reflexive.
You asked about Pinter's quote it is found in chapter 4 page 87.
You can do much better than Conover for topology.
Look at Principles of Topology by Fred Croom.

[imath][/imath][imath][/imath][imath][/imath]
 
Cratylus said:
I was only interested 1,2,5 . So in my post I stated them as 1,2,3
Click to expand...
You seem to be missing a lot of what we say.

First, do you not see that your #3 differs in two points from the #5 you showed? It is a different set, and a different relation! Please acknowledge this.

Second, you haven't shown me the answer given in your book, so that I can show you how you are misinterpreting that.

Cratylus said:
On this page it states ex 1.2 Z,R are directed sets,but they don tshow it https://users.math.msu.edu/users/banelson/teaching/920/nets.pdf

I wanted to see how to prove them.
Click to expand...
Third, you are now saying that your real question is not about that exercise at all, but yet a different fact, namely that R and Z (under [imath]\le[/imath], not <), are directed sets.

That is stated in the linked discussion without explanation, because it is obvious!

Definition 1.1. A directed set I is a set equipped with a binary relation ≤ that satisfies:​
(i) i ≤ i for all i ∈ I (reflexive);​
(ii) if i ≤ j and j ≤ k, then i ≤ k (transitive);​
(iii) for any i, j ∈ I there exists k ∈ I with i, j ≤ k (upper bound property).​
Typically reflexivity and transitivity are obvious, whereas the upper bound property may need to be​
justified.​

Just as it says, reflexivity and transitivity are obvious (that is, already familiar) for [imath]\le[/imath] on R and Z. The upper bound property should also be obvious: Given any two numbers, at least one of them is an upper bound! (Other relations on other sets, such as the subset relation on a set of sets, are not so simple.)

Cratylus said:
For reflexive relation, maybe there has to be some elemen that precedes x?
Click to expand...
It appears that you are genuinely confused about the meaning of reflexivity. Can you explain your thinking about this?

I can see how you might think "some element that precedes x" is required for the upper bound property (though it is not), but not why you would say this about reflexivity. Please say more.
 
Any element say x related to itself is reflexive. If R is a relation then (x,x) [imath]\in[/imath] R is reflexive
 
Cratylus said:
Any element say x related to itself is reflexive. If R is a relation then (x,x) [imath]\in[/imath] R is reflexive
Click to expand...
Your grammar here is wrong -- that is, your logic.

An element is not reflexive. An ordered pair is not reflexive. A relation can be reflexive.

What you presumably mean to say is,

IF EVERY element x IS related to itself, then THE RELATION is reflexive.​

And

If R is a relation AND FOR EVERY ELEMENT x, (x,x) [imath]\in[/imath] R, then R is reflexive.​

If you want to talk about math, you need to learn to say what you mean. Math at this level requires careful expression.

But how does this answer any of my questions? Am I right about #5? Does your book really say the answer is yes? Do you believe it is? And what did you mean about "some element that precedes x"?
 
Dr.Peterson said:
You seem to be missing a lot of what we say.

First, do you not see that your #3 differs in two points from the #5 you showed? It is a different set, and a different relation! Please acknowledge this.

Second, you haven't shown me the answer given in your book, so that I can show you how you are misinterpreting that.


Third, you are now saying that your real question is not about that exercise at all, but yet a different fact, namely that R and Z (under [imath]\le[/imath], not <), are directed sets.

That is stated in the linked discussion without explanation, because it is obvious!

Definition 1.1. A directed set I is a set equipped with a binary relation ≤ that satisfies:​
(i) i ≤ i for all i ∈ I (reflexive);​
(ii) if i ≤ j and j ≤ k, then i ≤ k (transitive);​
(iii) for any i, j ∈ I there exists k ∈ I with i, j ≤ k (upper bound property).​
Typically reflexivity and transitivity are obvious, whereas the upper bound property may need to be​
justified.​

Just as it says, reflexivity and transitivity are obvious (that is, already familiar) for [imath]\le[/imath] on R and Z. The upper bound property should also be obvious: Given any two numbers, at least one of them is an upper bound! (Other relations on other sets, such as the subset relation on a set of sets, are not so simple.)


It appears that you are genuinely confused about the meaning of reflexivity. Can you explain your thinking about this?

I can see how you might think "some element that precedes x" is required for the upper bound property (though it is not), but not why you would say this about reflexivity. Please say more.
Click to expand...
 
That is what I meant.
You are right 5 in the text is Z+ with <=
I put Z with < . Mine is strict order
 
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