different between codomain vs surjection

English

Etymology

co- +? domain

Pronunciation

• (US) IPA^(key): /?ko?.do??me?n/

Noun

codomain (plural codomains)

1. (mathematics, mathematical analysis) The target set into which a function is formally defined to map elements of its domain; the set denoted Y in the notation f : X ? Y.
   • 1994, Richard A. Holmgren, A First Course in Discrete Dynamical Systems, Springer, page 11,

     Definition 2.5. A function is onto if each element of the codomain has at least one element of the domain assigned to it. In other words, a function is onto if the range equals the codomain.

   • 2006, Robert L. Causey, Logic, Sets, and Recursion, 2nd Edition, Jones & Bartlett Learning, page 192,

     Once we have described ${\displaystyle f}$ as a function from ${\displaystyle A}$ to ${\displaystyle B}$, by convention we will call ${\displaystyle B}$ the codomain, even though other sets, of which ${\displaystyle B}$ is a subset, could have been used. […] If ${\displaystyle y}$ is an element of the codomain, then ${\displaystyle y\in {\mathit {Img}}(f,A)}$ iff there is some ${\displaystyle x}$ in the domain such that ${\displaystyle f}$ maps ${\displaystyle x}$ to ${\displaystyle y}$.

   • 2017, Alan Garfinkel, Jane Shevtsov, Yina Guo, Modeling Life: The Mathematics of Biological Systems, Springer, page 12,

     For example, the codomain of ${\displaystyle g(X)=X^{3}}$ consists of all real numbers. A function links each element in its domain to some element in its codomain. Each domain element is linked to exactly one codomain element.

Usage notes

The codomain always contains the image of the function (the actual set of points to which points of the domain are mapped), and can be larger if the function is not surjective.

The term range is often synonymous with codomain, but can also be used as a synonym for image.

Synonyms

• (target set of a function): range

Antonyms

• (target set of a function): domain

Translations

Further reading

• Domain of a function on Wikipedia.Wikipedia
• Image (mathematics) on Wikipedia.Wikipedia
• Range (mathematics) on Wikipedia.Wikipedia
• Injective function on Wikipedia.Wikipedia
• Surjective function on Wikipedia.Wikipedia
• Bijection on Wikipedia.Wikipedia
• Codomain on Wolfram MathWorld

Anagrams

• monoacid

English

Etymology

From French surjection, introduced by Nicolas Bourbaki in their treatise Éléments de mathématique.Ultimately borrowed from Latin superiecti? (“a throwing over or on; (fig.) an exaggeration, a hyperbole”).

Pronunciation

• IPA^(key): /s??(?).d??k.??n/

Noun

surjection (plural surjections)

1. (set theory) A function that is a many-to-one mapping; (formally) Any function ${\displaystyle f:X\rightarrow Y}$ for which for every ${\displaystyle y\in Y}$, there is at least one ${\displaystyle x\in X}$ such that ${\displaystyle f(x)=y}$.
   • 1992, Rowan Garnier, John Taylor, Discrete Mathematics for New Technology, Institute of Physics Publishing, page 220,

     In some special cases, however, the number of surjections ${\displaystyle A\rightarrow B}$ can be identified.

   • 1999, M. Pavaman Murthy, A survey of obstruction theory for projective modules of top rank, Tsit-Yuen Lam, Andy R. Magid (editors), Algebra, K-theory, Groups, and Education: On the Occasion of Hyman Bass's 65th Birthday, American Mathematical Society, page 168,

     Let ${\displaystyle J=\cap _{i}m_{i}}$ be the (irredundant) primary decomposition of ${\displaystyle J}$. We associate to the pair ${\displaystyle (J,\omega )}$ the element ${\displaystyle \textstyle \sum _{i}(m_{i},\omega _{i})\in G}$, where ${\displaystyle \omega _{i}}$ is the equivalence class of surjections from ${\displaystyle L/m_{i}L\oplus (A/m_{i})^{n-1}}$ to ${\displaystyle m_{i}/m_{i}^{2}}$ induced by ${\displaystyle \omega }$.

   • 2003, Gilles Pisier, Introduction to Operator Space Theory, Cambridge University Press, page 43,

     In Banach space theory, a mapping ${\displaystyle u:E\rightarrow F}$ (between Banach spaces) is called a metric surjection if it is onto and if the associated mapping from ${\displaystyle E/{\text{ker}}(u)}$ to ${\displaystyle F}$ is an isometric isomorphism. Moreover, by the classical open mapping theorem, ${\displaystyle u}$ is a surjection iff the associated mapping from ${\displaystyle E/{\text{ker}}(u)}$ to ${\displaystyle F}$ is an isomorphism.

Synonyms

• (function that is a many-to-one mapping): surjective function

Related terms

• bijection
• injection
• surject

Translations

References

French

Etymology

Formed after bijection and injection.

Pronunciation

• IPA^(key): /sy?.??k.sj??/

Noun

surjection f (plural surjections)

1. (set theory) surjection

Derived terms

• surjectif