Range (mathematics)

f

is a function from domain X to codomain Y. The yellow 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.

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 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 with 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

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

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.

Mathematical logic
General	Formal language Formation rule Formal system Deductive system Formal proof Formal semantics Well-formed formula Set Element Class Classical logic Axiom Natural deduction Rule of inference Relation Theorem Logical consequence Axiomatic system Type theory Symbol Syntax Theory
Traditional logic	Proposition Inference Argument Validity Cogency Syllogism Square of opposition Venn diagram
Propositional calculus Boolean logic	Boolean functions Propositional calculus Propositional formula Logical connectives Truth tables Many-valued logic
Predicate logic	First-order Quantifiers Predicate Second-order Monadic predicate calculus
Naive set theory	Set Empty set Element Enumeration Extensionality Finite set Infinite set Subset Power set Countable set Uncountable set Recursive set Domain Codomain Image Map Function Relation Ordered pair
Set theory	Foundations of mathematics Zermelo–Fraenkel set theory Axiom of choice General set theory Kripke–Platek set theory Von Neumann–Bernays–Gödel set theory Morse–Kelley set theory Tarski–Grothendieck set theory
Model theory	Model Interpretation Non-standard model Finite model theory Truth value Validity
Proof theory	Formal proof Deductive system Formal system Theorem Logical consequence Rule of inference Syntax
Computability theory	Recursion Recursive set Recursively enumerable set Decision problem Church–Turing thesis Computable function Primitive recursive function

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Range (mathematics)

Distinguishing between the two uses

Formal definition

See also

Notes

References