# Definition:Constant Mapping

## Definitions

A constant mapping or constant function is a mapping $f_c: S \to T$ defined as:

$c \in T: f_c: S \to T: \forall x \in S: \map {f_c} x = c$

That is, every element of $S$ is mapped to the same element $c$ in $T$.

In a certain sense, a constant mapping can be considered as a mapping which takes no arguments.

## Also known as

A constant mapping is of course also known as a constant function.

## Examples

### Constant Mappings on Set of Cardinality $3$

Let $X = \set {a, b, c}$.

Let $S = \set {f_a, f_b, f_c}$ be the constant mappings from $X$ to $X$.

The Cayley table for composition on $S$ is as follows:

$\begin{array}{c|cccc} \circ & f_a & f_b & f_c \\ \hline f_a & f_a & f_a & f_a \\ f_b & f_b & f_b & f_b \\ f_c & f_c & f_c & f_c \\ \end{array}$

As can be seen, there is no identity element, so $\struct {S, \circ}$ is not a group.

## Also see

• Results about constant mappings can be found here.