Let $E$ and $F$ be subsets of some set $S$, which may be finite or infinite. Suppose we are given a bijection $g\colon E\to F$. Is there an explicit way to construct from $g$ a bijection $h\colon S\setminus F\to S\setminus E$?
By explicit, I mean that it is not enough to show that $g$ exists (this is clear, since the cardinalities of $S\setminus F$ and $S\setminus E$ are equal).
