surjection

English

A surjection

Etymology

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

Pronunciation

• IPA^(key): /sɜː(ɹ)ˈd͡ʒɛk.ʃən/

• Audio (Southern England):

  (file)

Noun

surjection (plural surjections)

1. (set theory) A function for which every element of the codomain is mapped to by some element of the domain; (formally) Any function $f:X\rightarrow Y$ for which for every $y\in Y$, there is at least one $x\in X$ such that $f(x)=y$.
   • 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 $A\rightarrow B$ can be identified.

   • 1999, M. Pavaman Murthy, “A survey of obstruction theory for projective modules of top rank”, in 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 $J=\cap _{i}m_{i}$ be the (irredundant) primary decomposition of $J$. We associate to the pair $(J,\omega )$ the element $\textstyle \sum _{i}(m_{i},\omega _{i})\in G$, where $\omega _{i}$ is the equivalence class of surjections from $L/m_{i}L\oplus (A/m_{i})^{n-1}$ to $m_{i}/m_{i}^{2}$ induced by $\omega$.

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

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

Synonyms

• surjective function
• onto function

Related terms

• bijection
• injection
• surject

Translations

function that is a many-to-one mapping

• Catalan: funció exhaustiva f
• Chinese:

  Mandarin: 满射 (zh) (mǎnshè)

• Czech: surjekce f
• Estonian: sürjektsioon, surjection (et) f
• Finnish: surjektio (fi)
• French: surjection (fr) f
• German: Surjektion (de) f, surjektive Funktion f
• Greek: επίρριψη (el) f (epírripsi)
• Ido: surjekto
• Italian: suriezione (it) f
• Japanese: 全射 (ja) (zensha)
• Korean: 전사(全射) (ko) (jeonsa)
• Polish: suriekcja f
• Portuguese: sobrejeção f
• Romanian: surjecție f, funcție surjectivă f
• Russian: сюръе́кция (ru) f (sjurʺjékcija)
• Serbo-Croatian: surjekcija (sh) f
• Spanish: función sobreyectiva f
• Swedish: surjektion (sv) c

French

Etymology

Borrowing from Latin superiectiōnem (“a throwing over or on; (figuratively) an exaggeration, a hyperbole”). Compare injection, bijection, with the same second element but different prefixes.

Pronunciation

• IPA^(key): /syʁ.ʒɛk.sjɔ̃/

• Audio:

  (file)

Noun

surjection f (plural surjections)

1. (set theory) surjection

Derived terms

• surjectif