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:

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:


 * Surjection.png


 * $S$ 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:

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 $x \in T$