Definition:Codomain (Set Theory)/Mapping
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Definition
The codomain of a mapping $f: S \to T$ is the set $T$.
It can be denoted $\operatorname{Cdm} \left({f}\right)$.
Some sources, for example T.S. Blyth: Set Theory and Abstract Algebra (1975), also refer to the codomain as the arrival set.
A note on terminology
Some sources refer to the codomain of a mapping as its range.
However, other sources equate the term range with the image set.
Other sources brush the question aside by refraining from giving the codomain a name at all
As there exists significant ambiguity as to whether the range is to mean the codomain or image set, it is advised that the term range is not used.
The notation $\operatorname{Cdm} \left({f}\right)$ has not actually been found by this author anywhere in the literature. In fact, except in the field of category theory, no symbol for the concept of codomain has been found, despite extensive searching.
However, using $\operatorname{Cdm}$ to mean codomain is a useful enough shorthand to be worth coining. That is the approach which has been taken on this website.
Also see
Notes and References
- ↑ 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."
Sources
- T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 4$
- Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 20$
- H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): Appendix $\text{A}.3$
