Definition:Mapping/Definition 3
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Definition
Let $S$ and $T$ be sets.
A mapping $f$ from $S$ to $T$, denoted $f: S \to T$, is a relation $f = \struct {S, T, R}$, where $R \subseteq S \times T$, such that:
- $\forall \tuple {x_1, y_1}, \tuple {x_2, y_2} \in f: y_1 \ne y_2 \implies x_1 \ne x_2$
and
- $\forall x \in S: \exists y \in T: \tuple {x, y} \in R$
Notation
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 $\mathsf{Pr} \infty \mathsf{fWiki}$.
Also see
- Results about mappings can be found here.
Sources
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.10$: Functions: Definition $10.1$
- 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.1$: Functions: Ponderable $2.1.3$