I'm trying to find the error in a proof that yields a contradictory result, and I'm beginning to think that one of the definitions I start with is incorrect or self-contradictory. Is the following statement true?
"For any two distinct sets $A$ and $B$ and a function $f$ mapping from set $A$ to set $B$, if for any element $y$ in the range of $f$ there exists exactly one element $x$ in $A$ such that $f(x) = y$, but there exists at least one element $z$ in $B$ such that there is no element $x$ in $A$ for which $f(x) = z$, then set $B$ contains more elements than set $A$."
Thanks in advance!