help in deriving my equation (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.

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

[John]

I'm not sure I completely understand what you're asking here.

To be precise, your original relation [math]R[/math] isn't transitive.

 

You could probably recursively define your desired relation [math]R'[/math] as follows.

 

Given a relation [math]R_{1}[/math] on a set [math]S[/math]:

 

[math]\begin{array}{rcl}R_{n} & = & R_{n-1} \cup \{(a,b) \mid \exists x(aR_{n-1}x \wedge xR_{n-1}b) \}, \\ R' & = & R_{\min \{n \mid R_{n} = R_{n-1}\}}.\end{array}[/math]

 

There's probably a prettier way to express this, but that's the gist.

[mm84010]

Thanks John.