Mapping

Quick recap of definitions.

Table of contents

Basic definitions

Let $f$ be a function or mapping from a set $A$ to a set $B$ :

$f : X \to Y$

The set $X$ is called the domain of $f$ .

The set $Y$ is called the codomain of $f$ .

When $f (x_{i}) = y_{i}$ , we call $x_{i}$ the image of $y_{i}$ under $f$ .

The set of all images of $f$ is called the range of $f$ :

{f (x_{i}) ∣ x_{i} \in X}

Sometimes image and range are used interchangeably.

Let $f : X \to Y$ , $f$ is surjective if

$\forall y \in Y, \exists x \in X s . t . f (x) = y$

In other words, if the range of $f$ is equal to its codomain.

Also called onto mapping.

Let $f : X \to Y$ , $f$ is injective if

$\forall x_{1}, x_{2} \in X, f (x_{1}) = f (x_{2}) \Rightarrow x_{1} = x_{2}$

Also called one-to-one mapping.

Let $f : X \to Y$ , $f$ is bijective if it is both surjective and injective.

Also called one-to-one correspondence.

Let $f : X \to Y$ be a bijection, then there exists a inverse mapping $f^{- 1} : Y \to X$ such that

$\forall x \in X, f^{- 1} (f (x)) = x$

And this inverse mapping is also a bijection.