# Relations - Set Theory

• Apr 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 !!
• Apr 22nd 2006, 07:56 AM
Rebesques
Quote:

contains R^n.
...How is this relation defined? :confused:
• Apr 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:
• Apr 22nd 2006, 11:56 AM
rgep
If $\displaystyle R$ is a relation between sets $\displaystyle A$ and $\displaystyle B$: that is, $\displaystyle R \subseteq A \times B$, and similarly $\displaystyle S$ is a relation between $\displaystyle B$ and $\displaystyle C$, then the composition $\displaystyle R \circ S$ is the relation between $\displaystyle A$ and $\displaystyle C$ given by $\displaystyle \{ (a,c) \in A \times C : (a,b) \in R \mbox{ and } (b,c) \in S \mbox{ for some } b \in B \}$.

If $\displaystyle R$ is a relation between $\displaystyle A$ and itself, then it makes sense to define $\displaystyle R^2$ as $\displaystyle R \circ R$ and more generally $\displaystyle R^n = R \circ R^{n-1}$. Of course it turns out that composition is associative so that all the possible ways of defining $\displaystyle R^n$ are the same.
• Apr 22nd 2006, 11:07 PM
WeeG
thanks a lot !!!!

I am impressed !!
:)

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

Alright Weeg, can you tackle it now?