Let $R:A\to B$ a relation. For each $a \in A$ define $R(a)=\left\lbrace x\in A:aRx \right\rbrace$. Prove that, if $A$ and $B$ are finite sets, then: $$Car(R)=\sum_{a\in A} Car(R(a)).$$
Until now, I suppose $Car(A)=n$ and $Car(B)=m$ for some integers $n,m$, so $Car(R)=nm$, that is why I think it must be proven that $$\sum_{a\in A} Car(R(a)) = nm.$$ But I don't know how to proceed. I really appreciate some hints.