# Thread: Prove or disprove that R^4 is a transitive relation

1. ## Prove or disprove that R^4 is a transitive relation

Let $\displaystyle R$ be a transitive relation defined on a set $\displaystyle B$. Prove or disprove that $\displaystyle R^4$ is a transitive relation.

I'm inclined to believe that it's true, but I can't be sure. I at least know this: A relation $\displaystyle R$ is transitive on a set $\displaystyle A$ if whenever $\displaystyle (a,b)\in R$ and $\displaystyle (b,c)\in R$, then $\displaystyle (a,c)\in R$, $\displaystyle \forall a,b,c\in R$

2. Suppose that $\displaystyle R$ is transitive on $\displaystyle A$.
Suppose that $\displaystyle \left[ {\left( {a,b} \right) \in R \circ R} \right] \wedge \left[ {\left( {b,c} \right) \in R \circ R} \right]$.
By definition $\displaystyle \left( {\exists x \in A} \right)\left[ {(a,x) \in R \wedge (x,b) \in R} \right] \wedge \left( {\exists y \in A} \right)\left[ {(b,y) \in R \wedge (y,c) \in R} \right]$.
But that means that $\displaystyle \left[ {(a,b) \in R \wedge (b,c) \in R} \right]$ because $\displaystyle R$is transitive.
So by way of $\displaystyle b$ we have $\displaystyle {\left(a,c \right) \in R \circ R}$.
$\displaystyle R \circ R$ is transitive.
Can you extend that to $\displaystyle R^4?$