Definition:Mapping/Definition 4

Definition

Let $S$ and $T$ be sets.

A mapping from $S$ to $T$ is a relation on $S \times T$ which is:

$(1): \quad$ Many-to-one

$(2): \quad$ Left-total, that is, defined for all elements in $S$.

Let $f$ be a mapping.

This is usually denoted $f: S \to T$, which is interpreted to mean:

$f$ is a mapping with domain $S$ and codomain $T$

$f$ is a mapping of (or from) $S$ to (or into) $T$

$f$ maps $S$ to (or into) $T$.

The notation $S \stackrel f {\longrightarrow} T$ is also seen.

For $x \in S, y \in T$, the usual notation is:

$f: S \to T: \map f s = y$

where $\map f s = y$ is interpreted to mean $\tuple {x, y} \in f$.

It is read $f$ of $x$ equals $y$.

This is the preferred notation on $\mathsf{Pr} \infty \mathsf{fWiki}$.

1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 4$. Relations; functional relations; mappings
2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 2$: Functions