# Definition:Class Mapping

## Definition

Let $S$ and $T$ be classes.

A **class mapping $f$ from $S$ to $T$**, denoted $f: S \to T$, is a class relation $f = \mathcal R \subseteq S \times T$ such that:

- $\forall x \in S: \forall y_1, y_2 \in T: \left({x, y_1}\right) \in f \land \left({x, y_2}\right) \in f \implies y_1 = y_2$

and

- $\forall x \in S: \exists y \in T: \left({x, y}\right) \in f$

Usually, $\left({x, y}\right) \in f$ is instead written as:

- $f \left({x}\right) = y$

This is the preferred notation.

### Image and Preimage

Let $A \subseteq S$ and $B \subseteq T$.

Then **the image of $A$ under $f$** is defined as:

\(\displaystyle f \left({A}\right)\) | \(=\) | \(\displaystyle \left\{ {y \in B: \exists x \in A: f \left({x}\right) = y}\right\}\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \left\{ {f \left({x}\right): x \in A}\right\}\) | $\quad$ | $\quad$ |

Also, **the preimage of $B$ under $f$** is defined as:

- $f^{-1} \left({B}\right) = \left\{{x \in S: f \left({x}\right) \in B}\right\}$

## Also see

- Results about
**class mappings**can be found here.