need help in deriving my formula (Set theory)

[mm84010]

Hi,


I have a transitive relation and wana build a complete set of pairs that reflect all (direct/indirect) relations among the pairs.


Ex.: suppose I have this relation R = { (1,2), (2,3), (3,5), (5,7), (3,4) }


I wana to produce this relation R oper R = { (1,2), (1,3), (1,4), (1,5), (1,7), (2,3), (2,4), (2,5), (2,7), (3,4), (3,5), (3,7), (5,7) }


I tried to use the composite operator (°), but I got this R U (R ° R) = { (1,2), (2,3), (3,5), (5,7), (3,4), (1,3), (2,4), (2,5), (3,7) } which is not complete. In this case I need a loop operator until all pairs are restored.


Is there an operator that I can used to reflect that?


Thanks in advance.

 

[pka]

I have a transitive relation and want to build a complete set of pairs that reflect all (direct/indirect) relations among the pairs.
Ex.: suppose I have this relation \(\displaystyle R = \{ (1,2), (2,3), (3,5), (5,7), (3,4) \}\)

I do not understand what "want to build a complete set of pairs that reflect all (direct/indirect) relations among the pairs" means.

In any case you seem to not understand composition.
\(\displaystyle R\circ R=\{(1,3), (2,5),(2,4),(3,7)\}\)

If you want a symmetric relation that includes \(\displaystyle R\) that is simply \(\displaystyle R\cup R^{-1}.\)

 

[mm84010]

Thanks for passing by.

I do not understand what "want to build a complete set of pairs that reflect all (direct/indirect) relations among the pairs" means.

I want to go from this relation
R = { (1,2), (2,3), (3,5), (5,7), (3,4) }
to this one
R` = { (1,2), (1,3), (1,4), (1,5), (1,7), (2,3), (2,4), (2,5), (2,7), (3,4), (3,5), (3,7), (5,7) }

In any case you seem to not understand composition.
\(\displaystyle R\circ R=\{(1,3), (2,5),(2,4),(3,7)\}\)

If you looked to my example, I used the composite operator THEN I used the union operator. So I do not get your point here...

If you want a symmetric relation that includes \(\displaystyle R\) that is simply \(\displaystyle R\cup R^{-1}.\)

No, it does not give my objective (R`)

 

[mm84010]

Hi,


I am seeking help to derive my formula in set theory.


I will explain my request through the following example:


suppose I have this transitive relation R = { (1,2), (2,3), (3,5), (5,7), (3,4) }


I mean by transitive that since
(1,2) ==> 1 > 2, and
(2,3) ==> 2 > 3
then I can conclude that
1 > 3 or (1,3)
Hence, I can add this pair explicitly to R.


I wana to add all these implicit relations to reach the final relation:
R` = { (1,2), (1,3), (1,4), (1,5), (1,7), (2,3), (2,4), (2,5), (2,7), (3,4), (3,5), (3,7), (5,7) }


Currently, here are the steps I used to build my case


s1 = R ° R = { (1,3), (2,4), (2,5), (3,7) } // composite operator


s2 = R U s1 = { (1,2), (2,3), (3,5), (5,7), (3,4), (1,3), (2,4), (2,5), (3,7) } // union operator


s1 = s2 ° R = { (1,3), (2,4), (2,5), (3,7), (1,5), (1,4), (2,7) }


s2 = s2 U s1 = { (1,2), (2,3), (3,5), (5,7), (3,4), (1,3), (2,4), (2,5), (3,7), (1,5), (1,4), (2,7) }


s1 = s2 ° R = { (1,3), (2,4), (2,5), (3,7), (1,5), (1,4), (2,7), (1,7) }


s2 = s2 U s1 = { (1,2), (2,3), (3,5), (5,7), (3,4), (1,3), (2,4), (2,5), (3,7), (1,5), (1,4), (2,7), (1,7) }


s1 = s2 ° R = { (1,3), (2,4), (2,5), (3,7), (1,5), (1,4), (2,7), (1,7) }


I stop when s1 produce the same previous relation, and my result is in s2.


I built a computer algorithm for that, but I want a math formula to report my work in formal way.


I donot know if there is an operator reflect the generation of all implicit indirect relations among the elements of set (I read three discrete math books and browsed several math pages), or there is a loop operator that reflect a recursiveness based on a condition (not number of occurrences).


Thanks in advance
Reference to papers/books/tutorials are appreciated

 

[daon2]

Hi,


I am seeking help to derive my formula in set theory.


I will explain my request through the following example:


suppose I have this transitive relation R = { (1,2), (2,3), (3,5), (5,7), (3,4) }


I mean by transitive that since
(1,2) ==> 1 > 2, and
(2,3) ==> 2 > 3
then I can conclude that
1 > 3 or (1,3)
Hence, I can add this pair explicitly to R.


I wana to add all these implicit relations to reach the final relation:
R` = { (1,2), (1,3), (1,4), (1,5), (1,7), (2,3), (2,4), (2,5), (2,7), (3,4), (3,5), (3,7), (5,7) }


Currently, here are the steps I used to build my case


s1 = R ° R = { (1,3), (2,4), (2,5), (3,7) } // composite operator


s2 = R U s1 = { (1,2), (2,3), (3,5), (5,7), (3,4), (1,3), (2,4), (2,5), (3,7) } // union operator


s1 = s2 ° R = { (1,3), (2,4), (2,5), (3,7), (1,5), (1,4), (2,7) }


s2 = s2 U s1 = { (1,2), (2,3), (3,5), (5,7), (3,4), (1,3), (2,4), (2,5), (3,7), (1,5), (1,4), (2,7) }


s1 = s2 ° R = { (1,3), (2,4), (2,5), (3,7), (1,5), (1,4), (2,7), (1,7) }


s2 = s2 U s1 = { (1,2), (2,3), (3,5), (5,7), (3,4), (1,3), (2,4), (2,5), (3,7), (1,5), (1,4), (2,7), (1,7) }


s1 = s2 ° R = { (1,3), (2,4), (2,5), (3,7), (1,5), (1,4), (2,7), (1,7) }


I stop when s1 produce the same previous relation, and my result is in s2.


I built a computer algorithm for that, but I want a math formula to report my work in formal way.


I donot know if there is an operator reflect the generation of all implicit indirect relations among the elements of set (I read three discrete math books and browsed several math pages), or there is a loop operator that reflect a recursiveness based on a condition (not number of occurrences).


Thanks in advance
Reference to papers/books/tutorials are appreciated

Several results in mathematics are obtained via algorithms. There may not be a nice formula to spit things out, and I expect if there were it would be quite complicated as the initial state of R can be ugly. If you can prove your algorithm works then that is all that is needed.

 

[pka]

I am seeking help to derive my formula in set theory.
I will explain my request through the following example:
suppose I have this transitive relation R = { (1,2), (2,3), (3,5), (5,7), (3,4) }
I mean by transitive that since
(1,2) ==> 1 > 2, and
(2,3) ==> 2 > 3
then I can conclude that
1 > 3 or (1,3)
Hence, I can add this pair explicitly to R.

No one with even a rudimentary knowledge of set theory uses the term transitive relation that way.
That explains my confusion about your post.

On the set \(\displaystyle A=\{1,2,3,4,5,6,7\}\) the relation \(\displaystyle R = \{ (1,2), (2,3), (3,5), (5,7), (3,4) \}\) is a subset of the upper triangular relation \(\displaystyle \mathcal{T}=\{(m,n) :\{m,n\}\subset A~\&~m<n\}\)

One can generate closure using two do/until nested loops.