Definition:Range

Definition

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

The range of $\mathcal R$, denoted is defined as one of two things, depending on the source.

It is usually denoted $\operatorname{Rng} \left({\mathcal R}\right)$ or $\operatorname{Ran} \left({\mathcal R}\right)$ (or the same all in lowercase).

Range as Codomain

The range of a relation $\mathcal R \subseteq S \times T$ can be defined as the set $T$.

As such, it is the same thing as the term codomain of $\mathcal R$.

Range as Image

The range of a relation $\mathcal R \subseteq S \times T$ can also be defined as:

$\operatorname{Rng} \left({\mathcal R}\right) = \left\{{t \in T: \exists s \in S: \left({s, t}\right) \in \mathcal R}\right\}$

Defined like this, it is the same as what the image of $\mathcal R$.

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 see

• Domain
• Codomain

• Image
• Preimage

Sources

Those that define $\operatorname{Rng}$ as image:

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 7$: Relations
• W.E. Deskins: Abstract Algebra (1964): $\S 1.3$
• Seth Warner: Modern Algebra (1965): $\S 1$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 0.3$
• A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis (1968): $\S 1.3$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 4$
• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 7.1$
• Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.6$
• H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): Appendix $\text{A}.3$

Those that define $\operatorname{Rng}$ as codomain:

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 3.2$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): $\text{I}$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 10$

Those which do not use the term at all, but use codomain and image instead:

• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 20$

Some sources brush the question aside by refraining from giving a name to this concept at all:

• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): 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."