Definition:Surjection

Definition

Let $S$ and $T$ be sets or classes.

Let $f: S \to T$ be a mapping from $S$ to $T$.

Definition 1

$f: S \to T$ is a surjection if and only if:

$\forall y \in T: \exists x \in \Dom f: \map f x = y$

That is, if and only if $f$ is right-total.

Definition 2

$f: S \to T$ is a surjection if and only if:

$f \sqbrk S = T$

or, in the language and notation of direct image mappings:

$\map {f^\to} S = T$

That is, $f$ is a surjection if and only if its image equals its codomain:

$\Img f = \Cdm f$

Graphical Depiction

The following diagram illustrates the mapping:

$f: S \to T$

where $S$ and $T$ are the finite sets:

 $\ds S$ $=$ $\ds \set {a, b, c, i, j, k}$ $\ds T$ $=$ $\ds \set {p, q, r, s}$

and $f$ is defined as:

$f = \set {\tuple {a, p}, \tuple {b, p}, \tuple {c, q}, \tuple {i, r}, \tuple {j, s}, \tuple {k, s} }$

Thus the images of each of the elements of $S$ under $f$ are:

 $\ds \map f a$ $=$ $\ds \map f b = p$ $\ds \map f c$ $=$ $\ds q$ $\ds \map f i$ $=$ $\ds r$ $\ds \map f j$ $=$ $\ds \map f k = s$ $S$ is the domain of $f$.
$T$ is the codomain of $f$.
$\set {p, q, r, s}$ is the image of $f$.

The preimages of each of the elements of $T$ under $f$ are:

 $\ds \map {f^{-1} } p$ $=$ $\ds \set {a, b}$ $\ds \map {f^{-1} } q$ $=$ $\ds \set c$ $\ds \map {f^{-1} } r$ $=$ $\ds \set i$ $\ds \map {f^{-1} } s$ $=$ $\ds \set {j, k}$

Also known as

The phrase $f$ is surjective is often used for $f$ is a surjection.

Authors who prefer to limit the jargon of mathematics tend to use the term an onto mapping for a surjection, and onto for surjective.

A mapping which is not surjective is thence described as into.

A surjection $f$ from $S$ to $T$ is sometimes denoted:

$f: S \twoheadrightarrow T$

to emphasize surjectivity.

Examples

Arbitrary Finite Set

Let $S$ and $T$ be sets such that:

 $\ds S$ $=$ $\ds \set {a, b, c}$ $\ds T$ $=$ $\ds \set {x, y}$

Let $f: S \to T$ be the mapping defined as:

 $\ds \map f a$ $=$ $\ds x$ $\ds \map f b$ $=$ $\ds x$ $\ds \map f c$ $=$ $\ds y$

Then $f$ is a surjection.

Negative Function on Integers

Let $f: \Z \to \Z$ be the mapping defined on the set of integers as:

$\forall x \in \Z: \map f x = -x$

Then $f$ is a surjection.

Doubling Function on Reals

Let $f: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map f x = 2 x$

Then $f$ is a surjection.

Floor Function of $\dfrac {x + 1} 2$ on $\Z$

Let $f: \Z \to \Z$ be the mapping defined on the set of integers as:

$\forall x \in \Z: \map f x = \floor {\dfrac {x + 1} 2}$

where $\floor {\, \cdot \,}$ denotes the floor function.

Then $f$ is a surjection, but not an injection.

$\map f x = \dfrac x 2$ for $x$ Even, $0$ for $x$ Odd

Let $f: \Z \to \Z$ be the mapping defined on the set of integers as:

$\forall x \in \Z: \map f x = \begin{cases} \dfrac x 2 & : x \text { even} \\ 0 & : x \text { odd} \end{cases}$

Then $f$ is a surjection.

Also see

• Results about surjections can be found here.

Technical Note

The $\LaTeX$ code for $f: S \twoheadrightarrow T$ is f: S \twoheadrightarrow T .