Range (mathematics)
In mathematics, and more specifically in naive set theory, the range of a function refers to either the codomain or the image of the function, depending upon usage. Modern usage almost always uses range to mean image.
The codomain of a function is some arbitrary super-set of image. In real analysis, it is the real numbers. In complex analysis, it is the complex numbers.
The image of a function is the set of all outputs of the function. The image is always a subset of the codomain.
Distinguishing between the two uses
As the term "range" can have different meanings, it is considered a good practice to define it the first time it is used in a textbook or article.
Older books, when they use the word "range", tend to use it to mean what is now called the codomain.^{[1]}^{[2]} More modern books, if they use the word "range" at all, generally use it to mean what is now called the image.^{[3]} To avoid any confusion, a number of modern books don't use the word "range" at all.^{[4]}
As an example of the two different usages, consider the function as it is used in real analysis, that is, as a function that inputs a real number and outputs its square. In this case, its codomain is the set of real numbers , but its image is the set of non-negative real numbers , since is never negative if is real. For this function, if we use "range" to mean codomain, it refers to . When we use "range" to mean image, it refers to .
As an example where the range equals the codomain, consider the function , which inputs a real number and outputs its double. For this function, the codomain and the image are the same (the function is a surjection), so the word range is unambiguous; it is the set of all real numbers.
Formal definition
When "range" is used to mean "codomain", the image of a function f is already implicitly defined. It is (by definition of image) the (maybe trivial) subset of the "range" which equals {y | there exists an x in the domain of f such that y = f(x)}.
When "range" is used to mean "image", the range of a function f is by definition {y | there exists an x in the domain of f such that y = f(x)}. In this case, the codomain of f must not be specified, because any codomain which contains this image as a (maybe trivial) subset will work.
In both cases, image f ⊆ range f ⊆ codomain f, with at least one of the containments being equality.
See also
Notes
References
- Childs (2009). A Concrete Introduction to Higher Algebra. Undergraduate Texts in Mathematics (3rd ed.). Springer. ISBN 978-0-387-74527-5. OCLC 173498962.
- Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). Wiley. ISBN 978-0-471-43334-7. OCLC 52559229.
- Hungerford, Thomas W. (1974). Algebra. Graduate Texts in Mathematics. 73. Springer. ISBN 0-387-90518-9. OCLC 703268.
- Rudin, Walter (1991). Functional Analysis (2nd ed.). McGraw Hill. ISBN 0-07-054236-8.