different between surjection vs bijection

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

English

Etymology

From French bijection, introduced by Nicolas Bourbaki in their treatise Éléments de mathématique.

Pronunciation

• IPA^(key): /ba?.d??k.??n/

Noun

bijection (plural bijections)

1. (set theory) A one-to-one correspondence, a function which is both a surjection and an injection.
   • 2002, Yves Nievergelt, Foundations of Logic and Mathematics, page 214,

     The present text has defined a set to be finite if and only if there exists a bijection onto a natural number, and infinite if and only if there does not exist any such bijection.

   • 2007, C. J. Date, Logic and Databases: The Roots of Relational Theory, page 167,

     Note in particular that a function is a bijection if and only if it's both an injection and a surjection.

   • 2013, William F. Basener, Topology and Its Applications, unnumbered page,

     The basic idea is that two sets A and B have the same cardinality if there is a bijection from A to B. Since the domain and range of the bijection is not relevant here, we often refer to a bijection from A to B as a bijection between the sets, or a one-to-one correspondence between the elements of the sets.

Synonyms

• (function that is both a surjection and an injection): one-to-one correspondence

Related terms

• injection
• surjection

Translations

Anagrams

• objicient

French

Pronunciation

• IPA^(key): /bi.??k.sj??/

Noun

bijection f (plural bijections)

1. (set theory) bijection