> [!NOTE] **Definition** (Preimage) Suppose that $X$ and $Y$ are sets. Suppose that $f: X \to Y$ is a [[Function|function]]. For any $B \subset Y$ its *preimage* under $f$ is given by $f^{-1} (B) = \{ x \in X \mid f(x) \in B \}.$ > # Applications - [[Kernel of Homomorphism of Groups]].