Definition:Range of Relation/Codomain
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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 $T$.
As such, it is the same thing as the term codomain 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
- 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 $2.1.1$