Definition:Range of Relation/Image

Definition

Let $\RR \subseteq S \times T$ be a relation, or (usually) a mapping (which is, of course, itself a relation).

The range of $\RR$ can 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 of $\RR$.

Warning

Because of the ambiguity in definition, it is advised that the term range not be used in this context at all.

Instead that the term Codomain or Image be used as appropriate.

This is the approach to be taken consistently in $\mathsf{Pr} \infty \mathsf{fWiki}$.

Also denoted as

Some sources use the notation $\map {\mathrm {Ran} } \RR$ for the range of a relation (or the same all in lowercase).

Some sources use $\map {\mathsf {Ran} } \RR$.

Some use $R_\RR$.

Also see

• Definition:Domain of Relation
• Definition:Codomain of Relation
• Definition:Image of Relation
• Definition:Preimage

Sources

• 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 $\text I$: 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: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $1$: Pairs, Relations, and Functions
• 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): Appendix $\text{A}.3$: Functions
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): function (map, mapping)
• 1999: András Hajnal and Peter Hamburger: Set Theory ... (previous) ... (next): $1$. Notation, Conventions: $10$: Definition $1.3$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): function (map, mapping)
• 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
• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 2.1$: Continuous and linear maps. Linear transformations

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