Definition:Range of Relation
This page is about Range of Relation. For other uses, see Range.
Definition
Let $\RR \subseteq S \times T$ be a relation, or (usually) a mapping (which is, of course, itself a relation).
The range of $\RR$, denoted is defined as one of two things, depending on the source.
On $\mathsf{Pr} \infty \mathsf{fWiki}$ it is denoted $\Rng \RR$, but this may non-standard.
Range as Codomain
The range of a relation $\RR \subseteq S \times T$ can be defined as the set $T$.
As such, it is the same thing as the term codomain of $\RR$.
Range as Image
The range of a relation $\RR \subseteq S \times T$ can also be defined as:
- $\Rng \RR = \set {t \in T: \exists s \in S: \tuple {s, t} \in \RR}$
Defined like this, it is the same as what is defined as the image set of $\RR$.
Beware
Because of the ambiguity in definition, it is often advised that the term range not be used in this context at all, but instead that the term Codomain or Image be used as appropriate.
Also denoted as
Some sources use the notation $\map {\mathrm {Ran} } \RR$ (or the same all in lowercase).
Some sources use $\map {\mathsf {Ran} } \RR$.
Some use $R_\RR$
Also see
- Definition:Domain (Set Theory)
- Definition:Codomain (Set Theory)
- Definition:Image (Set Theory)
- Definition:Preimage
Sources
Those that define $\Rng \RR$ as image:
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 7$: Relations
- 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 1.3$
- 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (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: E.J. Lemmon: Beginning Logic ... (previous) ... (next): $\S 4.5$: Properties of Relations
- 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): $\S 0.3$
- 1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $\S 1.3$: Functions and mappings. Images and preimages
- 1970: Arne Broman: Introduction to Partial Differential Equations ... (previous) ... (next): Chapter $1$: Fourier Series: $1.1$ Basic Concepts: $1.1.2$ Definitions
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.10$: Functions: Definition $10.1$
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 6.5$
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 4$. Relations; functional relations; mappings
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 7.1$
- 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.6$: Functions
- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): Appendix $\text{A}.3$: Functions
- 1999: András Hajnal and Peter Hamburger: Set Theory ... (previous) ... (next): $1$. Notation, Conventions: $10$: Definition $1.3$
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 8$ Relations
Those that define $\Rng \RR$ as codomain:
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 3.2$. Equality of mappings
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Functions
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: Mappings: $\S 10$
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 6$: Functions
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.3$: Mappings
- 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 $2.1.1$
Those which do not use the term at all, but use codomain and image instead:
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 2$: Sets and functions: Graphs and functions
- also offering up range as a synonym for both codomain and image
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 20$: Introduction: Remarks $\text{(g)}$
Some sources brush the question aside by refraining from giving a name to this concept at all:
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): Notation and Terminology:
- A map or function (the terms are used interchangeably) between sets $A, B$ is written $f: A \to B$.
- We call $A$ the domain of $f$, and we avoid calling $B$ anything.