Suppose $f$ is a function whose domain is $X$ and codomain is $Y$. It is given that $|X|>1$ and $|Y|>1$. No other information is known about $X$, $Y$ and $f$. Instruction: Write the number of a single correct option for the given statement S.

$\mathrm{S} =$ "There exists a unique $y$ in $Y$ such that for each $x$ in $X$ it is true that $f(x) = y$." [1 point]

Options:
\begin{enumerate}
  \item S is always true.
  \item S is always false.
  \item S is true if and only if $f$ is one-to-one.
  \item If S is true then $f$ is one-to-one but the converse is false.
  \item If $f$ is one-to-one then S is true but the converse is false.
  \item S is true if and only if $f$ is onto.
  \item If S is true then $f$ is onto but the converse is false.
  \item If $f$ is onto then S is true but the converse is false.
  \item S is true if and only if $f$ is a constant function.
  \item If S is true then $f$ is a constant function but the converse is false.
  \item If $f$ is a constant function then S is true but the converse is false.
  \item None of the above.
\end{enumerate}