surjection

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Surjective function

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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”).^[1]

• IPA^(key): /sɜː(ɹ)ˈd͡ʒɛk.ʃən/

• Audio (Southern England):

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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 ${\displaystyle f:X\rightarrow Y}$ 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”, 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 ${\displaystyle J=\cap _{i}m_{i}}$ 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.

• surjective function
• onto function

• bijection
• injection
• surject

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.

• IPA^(key): /syʁ.ʒɛk.sjɔ̃/

• Audio:

  (file)

surjection f (plural surjections)

1. (set theory) surjection

• surjectif