In Folland, there is a statement as follows:
Any mapping $f: X \to Y$ between two sets induces a mapping $f^{-1}: \mathcal{P}(Y) \to \mathcal{P}({X})$ (these are power sets) defined by $f^{-1}(E) = \{x \in X: f(x) \in E\}$ which preserves unions, intersections and complements.
Here is my question: why does it like this? I mean, $f$ is a mapping from $X \to Y$, why its inverse mapping induces a mapping between power sets?