Injections, Surjections and Bijections

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 $y=f(x)$:

$$x_1\ne x_2⟹f(x_1)\ne f(x_2)$$

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.

$$f^{-1}(f(x))=x$$

Definition: Involution

An involution is a function $f$ which is its own inverse.

$$f=f^{-1}$$

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.