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 $f$:

\[\{f(x_i) \mid x_i \in X\}\]

Sometimes image and range are used interchangeably.

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

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

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.