Injections#
Definition: Injection
A function \(f: X \to Y\) is called injective if for each \(y\) in the image of \(f\) there is only one \(x\in X\) such that \(y = f(x)\):
\[ x_1 \ne x_2 \implies f(x_1) \ne f(x_2) \]
Definition: Inverse Function
The inverse function of an injection \(f: X \to Y\) is the function \(f^{-1}: f(X) \to X\) which to each \(y \in Y\) assigns the \(x \in X\) for which \(y = f(x)\), i.e.
\[ f^{-1}(f(x)) = x \]
Surjections#
Definition: Surjection
A function \(f: X \to Y\) is called surjective if its image and codomain are equal, i.e. for each \(y \in Y\) there is at least one \(x \in X\) such that \(y = f(x)\).
Bijections#
Definition: Bijection
A function \(f: X \to Y\) is called bijective if it is both injective and surjective.
