“Range (mathematics)”的意思、由来-开放百科全书

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Range (mathematics)

释义

Distinguishing between the two uses
Formal definition
See also
Notes
References

{{Other uses|Range (disambiguation){{!}}Range}}

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.

Notes

1. ^Hungerford 1974, page 3.
2. ^Childs 1990, page 140.
3. ^Dummit and Foote 2004, page 2.
4. ^Rudin 1991, page 99.

References

{{Cite book

| first =
| last = Childs
| title = A Concrete Introduction to Higher Algebra
| series = Undergraduate Texts in Mathematics
| edition = 3rd
| publisher = Springer
| year = 2009
| isbn = 978-0-387-74527-5
| oclc = 173498962
}}

{{Cite book

| first1 = David S.
| last1 = Dummit
| first2 = Richard M.
| last2 = Foote
| title = Abstract Algebra
| edition = 3rd
| publisher = Wiley
| year = 2004
| isbn = 978-0-471-43334-7
| oclc = 52559229
}}

{{Cite book

| first = Thomas W.
| last = Hungerford
| title = Algebra
| publisher = Springer
| series = Graduate Texts in Mathematics
| volume = 73
| year = 1974
| isbn = 0-387-90518-9
| oclc = 703268
}}

{{Cite book

| first = Walter
| last = Rudin
| title = Functional Analysis
| edition = 2nd
| publisher = McGraw Hill
| year = 1991
| isbn = 0-07-054236-8
}}{{Mathematical logic}}{{DEFAULTSORT:Range (Mathematics)}}

2 : Functions and mappings|Basic concepts in set theory

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Distinguishing between the two uses

Formal definition

See also

Notes

References