During one of my recent tests, I was given the following problem: "Let the relation $R$ be defined on all finite sets so that $ARB$ if and only if there exits a bijection from $A$ to $B$. Verify that $R$ is a transitive relation." I tried to go about this with the following proof, which was marked wrong:
To be transitive, the following must be true of a relation: if $ARB$ and $BRC$, then $ARC$. Consider three sets $A, B$, and $C$ such that $ARC$ and $BRC$. We know that $n(A) = n(B)$, and that $n(B) = n(C)$, as two sets form a bijection if and only if they have the same number of elements. Substituting $n(A) = n(B)$ into $n(B) = n(C)$, we obtain $n(A) = n(C)$. As the number of elements in $A$ and $C$ are the same, there exits a bijection between $A$ and $C$, and $ARC$ by the definition of $R$. Since $ARC$ if $ARB$ and $BRC$, $R$ is transitive by the definition of transitive.
I realize that there are better ways to go about proving this, but I don't understand why it is wrong. I was wondering if anyone would be able to tell me what is incorrect about my answer.