Definition:Mapping/Class Theory

Definition

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

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

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.

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.