MHB Proving Binary Relations: R & S

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The discussion focuses on proving the properties of binary relations R and S. Relation R, defined on natural numbers, is reflexive but not symmetric, antisymmetric, or transitive. Relation S, defined on integers and half-integers, is reflexive and symmetric, but its antisymmetry and transitivity require further analysis. Participants clarify the definitions of reflexivity, symmetry, and the implications of inequalities in relation R. The conversation also addresses the proper use of LaTeX for mathematical expressions.
andrew1
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Hi,

I'm currently stuck on a few questions regarding binary relations as I'm unsure on how to prove their properties.

R is defined on N by aRb if and only if a <= b and b <= a+5

Is R reflexive, symmetric, antisymmetric, transitive?


S is defined on Z (union) {x + 1/2 : x is an element of all integers} by aSb if and only if a - b is an even integer.

Is S reflexive, symmetric, antisymmetric, transitive?
 
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Is $\le$ reflexive? Why? What can you say about reflexivity of $R$?

Also why do you use LaTeX commands but do not surround formulas with dollar signs or [math] tags?
 
Sorry about the latex commands thing, not sure how it works :S

In regards to R, would a + 5 be treated as just a? therefore it isn't reflexive?
 
andrew said:
Sorry about the latex commands thing, not sure how it works
Even after I said that you can surround formulas with dollar signs?

andrew said:
In regards to R, would a + 5 be treated as just a? therefore it isn't reflexive?
$a+5$ is almost never treated as $a$; these are two different numbers. I suspect you don't know what "reflexive" means. Please review your textbook or Wikipedia and feel free to ask questions if you don't understand the explanation.
 
reflexivity is just aRa so I guess it would be reflexive since if I use any natural number for a it will be related to itself
 
andrew said:
reflexivity is just aRa
Yes. More accurately speaking, $R$ is reflexive if $aRa$ holds for all $a$.

andrew said:
so I guess it would be reflexive since if I use any natural number for a it will be related to itself
Yes. How to explain this in more detail? If $a\in\Bbb N$, then by definition of $R$, $aRa$ means $a\le a\le a+5$, which is true for any $a$. Now try proving reflexivity of $S$.

Concerning symmetry of $R$, what happens when $a<b$?

By the way, you can click the "Reply With Quote" button to see how formulas are typed.
 
Evgeny.Makarov said:
Yes. More accurately speaking, $R$ is reflexive if $aRa$ holds for all $a$.

Yes. How to explain this in more detail? If $a\in\Bbb N$, then by definition of $R$, $aRa$ means $a\le a\le a+5$, which is true for any $a$. Now try proving reflexivity of $S$.

Concerning symmetry of $R$, what happens when $a<b$?

By the way, you can click the "Reply With Quote" button to see how formulas are typed.

Well when $a<b$ won't be related to a since 4 < 5 for example doesn't mean 5 < 4, but now the anti-symmetric property part of the question confuses me because if I take a as 4 again and b as 5, 4 $\ne$ 5. Would it also be sufficient to say that the relation isn't transitive as well since 1R5 for example and 5R11 but 1 isn't related to 11?
 
andrew said:
Well when $a<b$ won't be related to a since 4 < 5 for example doesn't mean 5 < 4
Let's say it precisely. The expression $a<b$ cannot be related to $a$ because $a<b$ is not a number, and $R$ is a relation on numbers. Only a number can be related to another number via $R$. Further, 4 < 5 does not indeed imply 5 < 4, but so what? What does it have to do with $R$, which involved two inequalities (and both of them non-strict)? You have the right idea, but you need to say it precisely.

andrew said:
but now the anti-symmetric property part of the question confuses me because if I take a as 4 again and b as 5, 4 $\ne$ 5.
But is it the case that $4R5$ and $5R4$? If not, then this example does not violate antisymmetry.

andrew said:
Would it also be sufficient to say that the relation isn't transitive as well since 1R5 for example and 5R11 but 1 isn't related to 11?
It is not the case that $5R11$, but you are right that $R$ is not transitive.
 
This is what I suggest:

Choose $a$ and $b$ from the set $\{1,2,3,4,5,6,7,8,9\}$.

This isn't all of $\Bbb N$, but it will give you an idea of which number-pairs are related.

For example, do we have $1R9$? Let's check:

Is $1 \leq 9$? Yes, so we passed the first test.

Is $9 \leq 1+5$? no, so we failed the second test, and thus the pair $(1,9)$ is not in $R$.

You might try to eliminate certain "kinds" of pairs at one fell swoop. For example, if $a > b$, can we have $aRb$?
 
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