# Definition:Surjection/Graphical Depiction

## Diagrammatic Presentation of Surjection on Finite Set

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:

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

Note that $f$ is surjective but not injective:

$\map {f^{-1} } p$ is not a singleton: $\map f a = \map f b$
$\map {f^{-1} } x \ne \O$ for all $x \in T$