Definition:Class Mapping

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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.