Binary relations with text

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arhzz1

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Hello! I'm having a hard time solving these binary relations. They look like this;

Check the binary relations for (Set out of all the Lands): reflexivity,symmetry,antisymmetry and trasitivity

[MATH]xRy \Leftrightarrow x~ and~ y ~ have ~ the ~ same~ border[/MATH][MATH]xRy \Leftrightarrow The~ capital~ of ~ x lies ~ further ~ north[/MATH]
Now I'm not sure how to apply the rules of symmetry and antisymmetry to these.I've tried for the first one and I'm not sure if its right (feels like shooting fish in a barrel)

So for the first one I've got that is reflexiv,symmetric and transinitiv. Is this correct? How do I solve these type of problems.

Thank you!
 
The first one does not have the transitive property. If location A is directly north of location B which is directly north of location C then surely ARB and BRC but ARC
 
Jomo said:
By using the definitions.
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I've tried that but it just does not make sense to me. For example to check it is reflexive;

"Every element is related to itself". Now how do I apply this to borders. Are borders related to themselves? Also I have x and y, am I susposed to check for each ?
 
arhzz1 said:
I've tried that but it just does not make sense to me. For example to check it is reflexive;

"Every element is related to itself". Now how do I apply this to borders. Are borders related to themselves? Also I have x and y, am I susposed to check for each ?
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Just ask yourself if x and x have the same border? There is no reason to now ask yourself if y and y have the same border
 
Jomo said:
Just ask yourself if x and x have the same border? There is no reason to now ask yourself if y and y have the same border
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Well, they have the same border. I'll give these a try.
 
arhzz1 said:
Hello! I'm having a hard time solving these binary relations. They look like this;

Check the binary relations for (Set out of all the Lands): reflexivity,symmetry,antisymmetry and trasitivity

[MATH]xRy \Leftrightarrow x~ and~ y ~ have ~ the ~ same~ border[/MATH][MATH]xRy \Leftrightarrow The~ capital~ of ~ x lies ~ further ~ north[/MATH]
Now I'm not sure how to apply the rules of symmetry and antisymmetry to these.I've tried for the first one and I'm not sure if its right (feels like shooting fish in a barrel)

So for the first one I've got that is reflexiv,symmetric and transinitiv. Is this correct? How do I solve these type of problems.

Thank you!
Click to expand...
I don't like the wording of the first; what does "have the same border" mean??

If two countries have exactly the same border, then they are the same country. That is one possible interpretation.

Or it might be taken to mean that two countries have a common border (that is, they are adjacent along some part of their borders); then they are neighboring countries. That is another possible interpretation, though it requires twisting the English a little; and it would not make much sense that a country is its own neighbor, so it is unclear whether to consider this reflexive.

It is not uncommon for questions like this to be a little awkward; that is commonly a matter of whether the relation has been defined clearly enough. You need to choose an interpretation and write it out as clearly as you can before proceeding. Once you've done this, you have your fish in a barrel and can start shooting -- it will often be that simple!
 
I am rethinking my 1st comment. Usually it is said that xRy if x and y share a common border. However your problem clearly states xRy if they have the same border.

Now how can it happen that say two counties have the same border? Answer this question and you are 99% done.
 
Jomo said:
I am rethinking my 1st comment. Usually it is said that xRy if x and y share a common border. However your problem clearly states xRy if they have the same border.

Now how can it happen that say two counties have the same border? Answer this question and you are 99% done.
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Yea I've talked to my professor about this not being destinct enough and he rephased the question saying do xand y share a common border.
 
Well, if x and y share a common border, do y and x share a common border?

Fish in a barrel if the problem is stated correctly.
 
The USA & Mexico share a border; the USA & Canada share a border. Is that transitive?
 
Well,that is a great question. Could you maybe tell me on this example how to check particullarly for reflexion and how for symmetry, because they are very similar to me.Also can a relation be symmetric and asymmetric?
 
arhzz1 said:
Well,that is a great question. Could you maybe tell me on this example how to check particullarly for reflexion and how for symmetry, because they are very similar to me.Also can a relation be symmetric and asymmetric?
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Surely every country has the same border as itself. Country \(A\) has a border with country \(A\).
If country \(A\) has a border with country \(B\) then country \(B\) has a border with country \(A\).
The relation is both reflexive & symmetric.
 
pka said:
Surely every country has the same border as itself. Country \(A\) has a border with country \(A\).
If country \(A\) has a border with country \(B\) then country \(B\) has a border with country \(A\).
The relation is both reflexive & symmetric.
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Can you try and explain the diffrence between symmetry and antisymmetry. The way I see it symmetry is "If country A has a border with country B then counbry B has a border with country A". And antisymmetry "If country A has a border with country B AND if country B has a border with countr A then they must have the same border". Am I interpreting this wrong because it would seem that this is also an antisymmtrtic relation? But can this be true, can a relation be symmetric and antisymmtric at the same time
 
arhzz1 said:
Can you try and explain the diffrence between symmetry and antisymmetry. The way I see it symmetry is "If country A has a border with country B then counbry B has a border with country A". And antisymmetry "If country A has a border with country B AND if country B has a border with countr A then they must have the same border". Am I interpreting this wrong because it would seem that this is also an antisymmtrtic relation? But can this be true, can a relation be symmetric and antisymmtric at the same time
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Let's check the definition:

A relation R on a set S is antisymmetric provided that distinct elements are never both related to one another. In other words xRy and yRx together imply that x=y.​

So you're close, but the conclusion is not "they must have the same border", but "they must be the same country". That's very different. (The definition of antisymmetry can't "reach inside" the definition of the relation and deal with which border it is; it only deals with whether they are the related at all.)

Is this true of the relation we're discussing? Clearly not. Antisymmetry and symmetry can't coexist unless nothing is related to anything but itself; the former says that no distinct elements can be related in both directions, while the latter says that pairs are always related in both directions, if they are related at all. Wikipedia puts it that way:

In mathematics, a homogeneous relation R on set X is antisymmetric if there is no pair of distinct elements of X each of which is related by R to the other. More formally, R is antisymmetric precisely if for all a and b in X​
​
if R(a, b) with a ≠ b, then R(b, a) must not hold,​
​
or, equivalently,​
​
if R(a, b) and R(b, a), then a = b.​
 
Dr.Peterson said:
Let's check the definition:

A relation R on a set S is antisymmetric provided that distinct elements are never both related to one another. In other words xRy and yRx together imply that x=y.​

So you're close, but the conclusion is not "they must have the same border", but "they must be the same country". That's very different. (The definition of antisymmetry can't "reach inside" the definition of the relation and deal with which border it is; it only deals with whether they are the related at all.)

Is this true of the relation we're discussing? Clearly not. Antisymmetry and symmetry can't coexist unless nothing is related to anything but itself; the former says that no distinct elements can be related in both directions, while the latter says that pairs are always related in both directions, if they are related at all. Wikipedia puts it that way:

In mathematics, a homogeneous relation R on set X is antisymmetric if there is no pair of distinct elements of X each of which is related by R to the other. More formally, R is antisymmetric precisely if for all a and b in X​
​
if R(a, b) with a ≠ b, then R(b, a) must not hold,​
​
or, equivalently,​
​
if R(a, b) and R(b, a), then a = b.​
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Ah now I see, thank you for the great answer
 
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