Definition:Mapping

A mapping (or a map) is a special kind of binary relation which relates a given element of one set to one element of another.

A mapping $$f$$ from $$S$$ to $$T$$ (or "on $$S$$ into $$T$$") is a relation $$f: S \times T$$ such that:

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

and

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

In the context of numbers, a mapping is usually referred to as a function. In some contexts, the term transformation is often used. The term operator is also seen.

That is, a mapping is a many-to-one relation which is defined for all elements in the domain.

A mapping $$f \subseteq S \times T$$ is defined at $$x \in S$$ iff:

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

If:

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

then $$f$$ is not defined at $$x$$, and indeed, $$f$$ is not technically a mapping at all.