Definition:Mapping/Definition 2
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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 $f = \struct {S, T, G}$, where $G \subseteq S \times T$, such that:
- $\forall x \in S: \forall y_1, y_2 \in T: \tuple {x, y_1} \in G \land \tuple {x, y_2} \in G \implies y_1 = y_2$
and
- $\forall x \in S: \exists y \in T: \tuple {x, y} \in G$
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
- 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Introduction $\S 2$: Product sets, mappings
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Functions
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 8$: Functions
- 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 1.3$: Definition $1.8$
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 1$: The Language of Set Theory
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.4$
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Functions
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.10$: Functions: Definition $10.1$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 20$: Introduction: Remarks $\text {(k)}$
- 2011: Robert G. Bartle and Donald R. Sherbert: Introduction to Real Analysis (4th ed.) ... (previous) ... (next): $\S 1.1$: Sets and Functions