# Definition:Mapping/Class Theory

## Definition

Let $V$ be a basic universe.

Let $A \subseteq V$ and $B \subseteq V$ be classes.

In the context of class theory, a **mapping from $A$ into $B$** is a relation $f \subseteq A \times B$ such that:

- $\forall x \in A: \exists! y \in B: \tuple {x, y} \in f$

That is:

- $\forall x \in A: \forall y_1, y_2 \in B: \tuple {x, y_1} \in f \land \tuple {x, y_2} \in f \implies y_1 = y_2$

and

- $\forall x \in A: \exists y \in B: \tuple {x, y} \in f$

## Also known as

Words which are often used to mean the same thing as **mapping** include:

**transformation**(particularly in the context of self-maps)**operator**or**operation****function**(usually in the context of numbers)**map**(but this term is discouraged, as the term is also used by some writers for**planar graph**).

Some sources introduce the concept with informal words such as **rule** or **idea** or **mathematical notion**.

Sources which define a **mapping (function)** to be only a many-to-one relation refer to a **mapping (function)** as a **total mapping (total function)**.

Some use the term **single-valued relation**.

Sources which go into analysis of multifunctions often refer to a conventional **mapping** as:

- a
**single-valued mapping**or**single-valued function** - a
**many-to-one mapping**,**many-to-one transformation**, or**many-to-one correspondence**, and so on.

The wording can vary, for example: **many-one** can be seen for **many-to-one**.

A **mapping $f$ from $S$ to $T$** is also described as a **mapping on $S$ into $T$**.

In the context of class theory, a **mapping** is often seen referred to as a **class mapping**.

## Also see

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

## Sources

- 2002: Thomas Jech:
*Set Theory*(3rd ed.) ... (previous) ... (next): Chapter $1$: Power Set - 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 9$ Functions