Let \(f:X\to Y\) be a map of sets. Give, with proof, necessary and sufficient conditions for \[ f^{-1}(f(S))=S \] for every subset \(S\subseteq X\).
Solution
The condition is that \(f\) is injective.
For every map and every \(S\subseteq X\), one has \(S\subseteq f^{-1}(f(S))\). If \(f\) is injective and \(x\in f^{-1}(f(S))\), then \(f(x)=f(s)\) for some \(s\in S\). Injectivity gives \(x=s\), so \(x\in S\). Thus equality holds.
Conversely, suppose the equality holds for every \(S\subseteq X\). If \(f(a)=f(b)\), then \[ b\in f^{-1}(f(\{a\}))=\{a\}, \] so \(b=a\). Hence \(f\) is injective.