Definition:Mapping/Definition 1
Definition
Let $S$ and $T$ be sets.
A mapping from $S$ to $T$ is a binary relation on $S \times T$ which associates each element of $S$ with exactly one element of $T$.
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 $\mathsf{Pr} \infty \mathsf{fWiki}$.
Sometimes the brackets are omitted: $f x = y$, as seen in Allan Clark: Elements of Abstract Algebra, for example.
The notation $f: x \mapsto y$ is often seen, read $f$ maps, or sends, $x$ to $y$.
In the context of index families, the conventional notation $x_i$ is used to denote the value of the index $i$ under the indexing function $x$.
Thus $x_i$ means the same thing as $\map x i$.
Some sources use this convention for the general mapping, thus:
- $f_x = y$
as remarked on in P.M. Cohn: Algebra Volume 1 (2nd ed.), for example.
Less common notational forms of $\map f s = y$ are:
- $x f = y$, as seen in Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts and 1968: Ian D. Macdonald: The Theory of Groups, for example
- $x^f = y$, as seen in Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts and John D. Dixon: Problems in Group Theory, for example
- This left-to-right style is referred to by some authors as the European convention.
John L. Kelley: General Topology provides a list of several different styles: $\tuple {f, x}$, $\tuple {x, f}$, $f x$, $x f$ and $\cdot f x$, and discusses the advantages and disadvantages of each.
The notation $\cdot f x$ is attributed to Anthony Perry Morse, and can be used to express complicated expressions without the need of parenthesis to avoid ambiguity. However, it appears not to have caught on.
Warning
The notation:
is an abuse of notation.
If $f: S \to T$ is a mapping, then $\map f x \in T$ for all $x \in S$.
Thus $\map f x$ is a mapping if and only if $\Img f$ is a set of mappings.
The point is that, as used here, $\map f x$ is not a mapping, but it is the image of $x$ under $f$.
Hence it is preferable not to talk about:
- the function $\cos x$
but instead should say:
- the function $\cos$
or:
- the function $x \mapsto \cos x$
although for the latter it would be better to also specify the domain and codomain.
Also defined as
Some sources do not place emphasis on the uniqueness of the element of $T$ that is being mapped to.
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
- 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 2.2$: Truth-Functions
- 1959: E.M. Patterson: Topology (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Topological Spaces: $\S 9$. Functions
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 8$: Functions
- 1963: Morris Tenenbaum and Harry Pollard: Ordinary Differential Equations ... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $2 \text B$: The Meaning of the Term Function of One Independent Variable
- 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 3$. Homomorphisms
- 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 1.3$: Definition $1.8$
- 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
- 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (next): Chapter $2$: Elements of Set Theory: Finite, Countable, and Uncountable Sets: $2.1$. Definition
- 1964: William K. Smith: Limits and Continuity ... (previous) ... (next): $\S 2.2$: Functions: Definition $1$
- 1965: Claude Berge and A. Ghouila-Houri: Programming, Games and Transportation Networks ... (previous) ... (next): $1$. Preliminary ideas; sets, vector spaces: $1.1$. Sets
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 3.1$. Mappings
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $1$: 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
- 1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $\S 1.3$: Functions and mappings. Images and preimages
- 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): Appendix: Elementary set and number theory
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: Mappings: $\S 10$
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 6.4$
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.10$: Functions: Definition $10.1$
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 2$: Sets and functions: Graphs and functions
- 1972: Murray R. Spiegel and R.W. Boxer: Theory and Problems of Statistics (SI ed.) ... (previous) ... (next): Chapter $1$: Functions
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 4$. Relations; functional relations; mappings
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 6$: Functions
- 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.4$: Functions
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 20$: Introduction
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.3$: Mappings
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.28$: Dirichlet ($\text {1805}$ – $\text {1859}$)
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $2$: Maps and relations on sets: Definition $2.1$
- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): $\S 1.5$: Semantics of Propositional Logic
- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): Appendix $\text{A}.3$: Functions
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 2$: Functions
- 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.1$: Functions: Definition $2.1.1$
- 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): Appendix $\text{A}$: Set Theory: Functions
- 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): Appendix $\text{A}.4$: Definition $\text{A}.23$