In

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, the codomain or set of destination of a function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range
Range may refer to:
Geography
* Range (geographic)A range, in geography, is a chain of hill
A hill is a landform
A landform is a natural or artificial feature of the solid surface of the Earth or other planetary body. Landforms together ...

is sometimes ambiguously used to refer to either the codomain or image
An image (from la, imago) is an artifact that depicts visual perception
Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...

of a function.
A codomain is part of a function if is defined as a triple where is called the ''domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Doma ...

'' of , its ''codomain'', and its ''graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discret ...

''. The set of all elements of the form , where ranges over the elements of the domain , is called the ''image
An image (from la, imago) is an artifact that depicts visual perception
Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...

'' of . The image of a function is a subset
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of its codomain so it might not coincide with it. Namely, a function that is not surjective
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

has elements in its codomain for which the equation does not have a solution.
A codomain is not part of a function if is defined as just a graph. For example in set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...

it is desirable to permit the domain of a function to be a proper class
Proper may refer to:
Mathematics
* Proper map
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and qu ...

, in which case there is formally no such thing as a triple . With such a definition functions do not have a codomain, although some authors still use it informally after introducing a function in the form ., p. 91 (quote 1
Quote is a hypernym of quotation, as the repetition or copy of a prior statement or thought. Quotation marks are punctuation marks that indicate a quotation. Both ''quotation'' and ''quotation marks'' are sometimes abbreviated as "quote(s)".
Co ...

quote 2
Quote is a hypernym
In linguistics
Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying and modeling them.
The traditional areas of ling ...

; , p. 8P. is an abbreviation or acronym that may refer to:
* Page (paper)
A page is one side of a leaf
A leaf (plural leaves) is the principal lateral appendage of the vascular plant plant stem, stem, usually borne above ground and specialized f ...

Mac Lane, in , p. 232P. is an abbreviation or acronym that may refer to:
* Page (paper)
A page is one side of a leaf
A leaf (plural leaves) is the principal lateral appendage of the vascular plant plant stem, stem, usually borne above ground and specialized f ...

, p. 91 , p. 89/ref>
Examples

For a function :$f\backslash colon\; \backslash mathbb\backslash rightarrow\backslash mathbb$ defined by : $f\backslash colon\backslash ,x\backslash mapsto\; x^2,$ or equivalently $f(x)\backslash \; =\backslash \; x^2,$ the codomain of is $\backslash textstyle\; \backslash mathbb\; R$, but does not map to any negative number. Thus the image of is the set $\backslash textstyle\; \backslash mathbb^+\_0$; i.e., the interval . An alternative function is defined thus: : $g\backslash colon\backslash mathbb\backslash rightarrow\backslash mathbb^+\_0$ : $g\backslash colon\backslash ,x\backslash mapsto\; x^2.$ While and map a given to the same number, they are not, in this view, the same function because they have different codomains. A third function can be defined to demonstrate why: : $h\backslash colon\backslash ,x\backslash mapsto\; \backslash sqrt\; x.$ The domain of cannot be $\backslash textstyle\; \backslash mathbb$ but can be defined to be $\backslash textstyle\; \backslash mathbb^+\_0$: : $h\backslash colon\backslash mathbb^+\_0\backslash rightarrow\backslash mathbb.$ The compositions are denoted : $h\; \backslash circ\; f,$ : $h\; \backslash circ\; g.$ On inspection, is not useful. It is true, unless defined otherwise, that the image of is not known; it is only known that it is a subset of $\backslash textstyle\; \backslash mathbb\; R$. For this reason, it is possible that , when composed with , might receive an argument for which no output is defined – negative numbers are not elements of the domain of , which is thesquare root function
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis) ...

.
Function composition therefore is a useful notion only when the ''codomain'' of the function on the right side of a composition (not its ''image'', which is a consequence of the function and could be unknown at the level of the composition) is a subset of the domain of the function on the left side.
The codomain affects whether a function is a surjection
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, in that the function is surjective if and only if its codomain equals its image. In the example, is a surjection while is not. The codomain does not affect whether a function is an injection
Injection or injected may refer to:
Science and technology
* Injection (medicine)
An injection (often referred to as a "shot" in US English, a "jab" in UK English, or a "jag" in Scottish English and Scots Language, Scots) is the act of adminis ...

.
A second example of the difference between codomain and image is demonstrated by the linear transformation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s between two vector space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s – in particular, all the linear transformations from $\backslash textstyle\; \backslash mathbb^2$ to itself, which can be represented by the matrices
Matrix or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...

with real coefficients. Each matrix represents a map with the domain $\backslash textstyle\; \backslash mathbb^2$ and codomain $\backslash textstyle\; \backslash mathbb^2$. However, the image is uncertain. Some transformations may have image equal to the whole codomain (in this case the matrices with rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking
A ranking is a relationship between a set of items such that, for any two items, the first is either "rank ...

) but many do not, instead mapping into some smaller subspace (the matrices with rank or ). Take for example the matrix given by
:$T\; =\; \backslash begin\; 1\; \&\; 0\; \backslash \backslash \; 1\; \&\; 0\; \backslash end$
which represents a linear transformation that maps the point to . The point is not in the image of , but is still in the codomain since linear transformations from $\backslash textstyle\; \backslash mathbb^2$ to $\backslash textstyle\; \backslash mathbb^2$ are of explicit relevance. Just like all matrices, represents a member of that set. Examining the differences between the image and codomain can often be useful for discovering properties of the function in question. For example, it can be concluded that does not have full rank since its image is smaller than the whole codomain.
See also

* *Notes

References

* * * * * * * {{Mathematical logic Functions and mappingsBasic concepts in set theory{{Commons
This category is for the foundational concepts of naive set theory, in terms of which contemporary mathematics is typically expressed.
Mathematical concepts ...