Definition of $n$-to-$1$ mapping. What is the definition of "$n$-to-$1$ mapping"?
Does an $n$-to-$1$ mapping mean to say that if $f$ is a function from $A$ to $B$, then for every $y\in R(f)$ there exists $n$ different elements in $A$ which maps to $y$ ?

Yes. To be more accurate, a map $f:X\to Y$ is $n$-to-$1$ iff $\forall y\in Y$ there are exactly $n$ different $x\in X$ such that $f(x)=y$. In other words, $\lvert f^{-1}(\{y\})\rvert=n$.
