different between injection vs surjection

English

Etymology

Borrowed from Middle French injection, from Latin iniectio. The mathematical sense is from French injection, introduced by Nicolas Bourbaki in their treatise Éléments de mathématique.

Pronunciation

• IPA^(key): /?n.?d??k.??n/
• Hyphenation: in?jec?tion
• Rhymes: -?k??n

Noun

injection (countable and uncountable, plural injections)

1. The act of injecting, or something that is injected.
2. A specimen prepared by injection.
3. (category theory) A morphism from either one of the two components of a coproduct to that coproduct.
4. (construction) The act of inserting materials like concrete grout or gravel by using high pressure pumps.
5. (figuratively) The supply of additional funding to a person or a business.

   The troubled business received a much-needed cash injection.

6. (mathematics) A relation on sets (X,Y) that associates each element of Y with at most one element of X.
7. (programming) The insertion of program code into an application, URL, hardware, etc.; especially when malicious or when the target is not designed for such insertion.

   a SQL injection exploit allowing a malicious user to modify a database query

8. (space science) The act of putting a spacecraft into a particular orbit, especially for changing a stable orbit into a transfer orbit, e.g. trans-lunar injection.
   • 2015, Henry L. Richter, America's Leap Into Space

     It had been determined that one of the whip turnstile antennas had broken off from Explorer 1 shortly after injection into orbit, so these were eliminated.

9. (set theory) A function that maps distinct x in the domain to distinct y in the codomain; formally, a f: X ? Y such that f(a) = f(b) implies a = b for any a, b in the domain.
10. (specifically, medicine) Something injected subcutaneously, intravenously, or intramuscularly by use of a syringe and a needle.
11. (steam engines) The act of throwing cold water into a condenser to produce a vacuum.
12. (steam engines) The cold water thrown into a condenser to produce a vacuum.

Hyponyms

Derived terms

Related terms

• inject

Translations

See also

References

• injection on Wikipedia.Wikipedia

French

Etymology

Borrowed from Latin iniecti?, iniecti?nem.

Pronunciation

• IPA^(key): /??.??k.sj??/

Noun

injection f (plural injections)

1. injection

Related terms

• injecter

Further reading

• “injection” in Trésor de la langue française informatisé (The Digitized Treasury of the French Language).

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