So I'm told that for a function $f:X\rightarrow Y$ to be surjective then
$$\forall y\in Y~,~\exists x\in X~,~f(x)=y,$$
so does this "$\exists$" imply more than one such $x$ can hit the same $y$? That is, am I right in thinking that
$$\forall y\in Y~,~\exists ! x\in X~,~f(x)=y$$
is an incorrect definition of surjectivity?