How can I write down this property of relations

In summary, the speaker is discussing a relation that is not only antisymmetric, but also has the property that if a relates to b and b relates to c, then c does not relate to a. They are trying to determine if this property holds for any string of related elements, without having to write it down forever. The speaker is specifically looking for a way to write this property down simply, but is struggling to do so. They mention that it may be similar to a partial ordering or an acyclic directed graph.
  • #1
V0ODO0CH1LD
278
0
If I have a relation which is not only antisymmetric (##aRb\rightarrow{}\neg{}bRa##) but it also has a property that ##aRb\land{}bRc\rightarrow{}\neg{}cRa##. How can I be sure that this property holds for any string like that? So that ##aRb\land{}bRc\land{}cRd\rightarrow{}\neg{}dRa## without having to write it down forever?

I though writing down ##aRb\land{}bRc\rightarrow{}\neg{}cRa## was enough, but with just that I can't prove that ##cRd\rightarrow{}\neg{}dRa##. How can I define this property? Does it exist already?
 
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  • #2
V0ODO0CH1LD said:
If I have a relation which is not only antisymmetric (##aRb\rightarrow{}\neg{}bRa##)

This is called asymmetric, not antisymmetric.

How can I be sure that this property holds for any string like that?

How do you know that what property holds for any string? You know that asymmetry and the other property hold because you've assumed it holds.

Do you somehow want to derive the property
[tex]aRb ~\wedge~bRc~\wedge~cRd~\rightarrow \neg dRa[/tex]
from the previous two? I think you can easily find examples of relations that are asymmetric and satisfy the second property and such that the above property doesn't hold.
 
  • #3
micromass said:
Do you somehow want to derive the property

[tex]aRb ~\wedge~bRc~\wedge~cRd~\rightarrow \neg dRa[/tex]

from the previous two? I think you can easily find examples of relations that are asymmetric and satisfy the second property and such that the above property doesn't hold.
Sorry, I'm having a hard time explaining this.. I want to create a relation for which asymmetry holds as well as this other property that if a relates to b relates to c relates to d and so on, then not only do they not relate the other way due to asymmetry but also that the last element in the "string" does not relate to any of the previous ones (eg d does not relate to b and a, and c does not relate a) which I don't think you can derive from asymmetry.

My question is: how can I write this property down simply?
 
  • #4
V0ODO0CH1LD said:
Sorry, I'm having a hard time explaining this.. I want to create a relation for which asymmetry holds as well as this other property that if a relates to b relates to c relates to d and so on, then not only do they not relate the other way due to asymmetry but also that the last element in the "string" does not relate to any of the previous ones (eg d does not relate to b and a, and c does not relate a) which I don't think you can derive from asymmetry.

My question is: how can I write this property down simply?

Perhaps what you are getting at is what is called a partial ordering. Here are a couple of links:

http://en.wikipedia.org/wiki/Partially_ordered_set

http://staff.scm.uws.edu.au/cgi-bin/cgiwrap/zhuhan/dmath/dm_readall.cgi?page=20&key
 
  • #5
LCKurtz said:
Perhaps what you are getting at is what is called a partial ordering.

Partial ordering satisfies reflexivity, antisymmetry and transitivity, that is not the relation that I am looking for.

The relation I am looking for satisfies the following conditions:
  • ¬(a~b) (anti-reflexitivity)
  • (a~b) then ¬(b~a) (asymmetry)
  • if (a~b) and (b~c) then ¬(c~a) (?)
In the third property, maybe (a~c) but that is not a requirement, the only requirement is that ¬(c~a). Not only that but also if (a~b) and (b~c) and (c~d) then ¬(d~a).
 
  • #6
V0ODO0CH1LD said:
In the third property, maybe (a~c) but that is not a requirement, the only requirement is that ¬(c~a). Not only that but also if (a~b) and (b~c) and (c~d) then ¬(d~a).

If you view the relation as a directed graph I think that what you are after is that it be "acyclic".
 

1. What is the definition of a property of relations?

A property of relations is a characteristic or attribute that describes the relationship between two or more items or entities.

2. How do I identify a property of relations?

To identify a property of relations, you must first understand the relationship between the items or entities. Look for patterns, rules, or commonalities that can be used to describe the relationship.

3. What are some common examples of properties of relations?

Some common examples of properties of relations include symmetry, transitivity, reflexivity, and antisymmetry. These properties can help describe how the items or entities in a relationship interact with each other.

4. Can properties of relations change over time?

Yes, properties of relations can change over time. As relationships between items or entities evolve, the properties that describe them may also change.

5. How can I effectively write down a property of relations?

To write down a property of relations, start by clearly defining the relationship between the items or entities. Use precise language and provide examples to illustrate the property. You may also use mathematical notation or diagrams to help explain the property.

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