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 $f = \RR \subseteq S \times T$ such that:


 * $\forall x \in S: \forall y_1, y_2 \in T: \tuple {x, y_1} \in f \land \tuple {x, y_2} \in f \implies y_1 = y_2$

and
 * $\forall x \in S: \exists y \in T: \tuple {x, y} \in f$

Usually, $\tuple {x, y} \in f$ is instead written as:
 * $\map f x = y$

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:

Also, the preimage of $B$ under $f$ is defined as:
 * $f^{-1} \sqbrk B = \set {x \in S: \map f x \in B}$

Also see

 * Definition:Class Injection
 * Definition:Class Surjection
 * Definition:Class Bijection