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 !!
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.