# Relations - Set Theory

• April 21st 2006, 11:52 AM
WeeG
Relations - Set Theory
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 !!
• April 22nd 2006, 07:56 AM
Rebesques
Quote:

contains R^n.
...How is this relation defined? :confused:
• April 22nd 2006, 09:44 AM
WeeG
Quote:

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....
:confused:
• April 22nd 2006, 11:56 AM
rgep
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.
• April 22nd 2006, 11:07 PM
WeeG
thanks a lot !!!!

I am impressed !!
:)

thanks !
• April 23rd 2006, 07:19 PM
Rebesques
That's the rgep we know... :)

Alright Weeg, can you tackle it now?