Relations - Set Theory

**WeeG** · April 21st 2006, 11:52 AM

hello everyone,

I have a problem with this question, I hope someone can help me out:

prove that a relations R over the set A is transitive if and only if for all n>=2 R contains R^n.

thanks !!

**Rebesques** · April 22nd 2006, 07:56 AM

contains R^n.

...How is this relation defined?

**WeeG** · April 22nd 2006, 09:44 AM

Originally Posted by WeeG

hello everyone,

R contains R^n.

I mean that R is included in R^n

I have no data except for that, it's a hard question....

**rgep** · April 22nd 2006, 11:56 AM

If $R$ is a relation between sets $A$ and $B$ : that is, $R \subseteq A \times B$ , and similarly $S$ is a relation between $B$ and $C$ , then the composition $R \circ S$ is the relation between $A$ and $C$ given by $\{ (a,c) \in A \times C : (a,b) \in R \mbox{ and } (b,c) \in S \mbox{ for some } b \in B \}$ .

If $R$ is a relation between $A$ and itself, then it makes sense to define $R^2$ as $R \circ R$ and more generally $R^n = R \circ R^{n-1}$ . Of course it turns out that composition is associative so that all the possible ways of defining $R^n$ are the same.

**WeeG** · April 22nd 2006, 11:07 PM

thanks a lot !!!!

I am impressed !!

thanks !

**Rebesques** · April 23rd 2006, 07:19 PM

That's the rgep we know...

Alright Weeg, can you tackle it now?

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