World Library  
Flag as Inappropriate
Email this Article

Range (mathematics)

Article Id: WHEBN0000275991
Reproduction Date:

Title: Range (mathematics)  
Author: World Heritage Encyclopedia
Language: English
Subject: Domain of a function, Function (mathematics), Codomain, Tarski–Grothendieck set theory, Interpretation (logic)
Collection: Basic Concepts in Set Theory, Functions and Mappings
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Range (mathematics)

f is a function from domain X to codomain Y. The smaller oval inside Y is the image of f. Sometimes "range" refers to the image and sometimes to the codomain.

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 set. 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.

Contents

  • Distinguishing between the two uses 1
  • Formal definition 2
  • See also 3
  • Notes 4
  • References 5

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 f(x) = x^2 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 \mathbb{R}, but its image is the set of non-negative real numbers \mathbb{R}^+, since x^2 is never negative if x is real. For this function, if we use "range" to mean codomain, it refers to \mathbb{R}. When we use "range" to mean image, it refers to \mathbb{R}^+.

As an example where the range equals the codomain, consider the function f(x) = 2x, 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 range of a function must be specified. It is often assumed to be the set of all real numbers, and {y | there exists an x in the domain of f such that y = f(x)} is called the image of f.

When "range" is used to mean "image", the range of a function f is {y | there exists an x in the domain of f such that y = f(x)}. In this case, the codomain of f must be specified, but is often assumed to be the set of all real numbers.

In both cases, image f ⊆ range f ⊆ codomain f, with at least one of the containments being equality.

See also

Notes

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

References

  • Childs (2009). A Concrete Introduction to Higher Algebra. Undergraduate Texts in Mathematics (3rd ed.). Springer.  
  • Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). Wiley.  
  • Hungerford, Thomas W. (1974). Algebra. Graduate Texts in Mathematics 73. Springer.  
  • Rudin, Walter (1991). Functional Analysis (2nd ed.). McGraw Hill.  
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 



Copyright © World Library Foundation. All rights reserved. eBooks from World eBook Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.