different between bijection vs morphism
bijection
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)
- (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.
- 2002, Yves Nievergelt, Foundations of Logic and Mathematics, page 214,
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)
- (set theory) bijection
bijection From the web:
- meaning of bijection
- what is bijection in graph theory
- what is bijection in combinatorics
- what natural bijection
- what is bijection meaning in hindi
- what is bijection in physics
- what are injections used for
- what is a bijection in discrete math
morphism
English
Etymology
Generalised from isomorphism, etc.
Noun
morphism (plural morphisms)
- (mathematics, category theory) (formally) An arrow in a category; (less formally) an abstraction that generalises a map from one mathematical object to another and is structure-preserving in a way that depends on the branch of mathematics from which it arises.
- 1982, Israel Program for Scientific Translations (translator), Lev J. Leifman (editor of translation), N. N. ?encov, Statistical Decision Rules and Optimal Inference, American Mathematical Society, Translations of Mathematical Monographs, Volume 53, page 50,
- 1° The composition of two morphisms is defined if and only if the final object of the first morphism is the initial object of the second. This composition is also a morphism, whose initial object is the initial object of the first morphism and whose final object is the final object of the second.
- 1992, Terrance Brown (translator), Gil Henriques, Chapter 13: Morphisms and Transformations in the Construction of Invariants, Terrance Brown (translator), Jean Piaget, Gil Henriques, Edgar Ascher (editors), Morphisms and Categories: Comparing and Transforming, page 198,
- In certain extreme cases in mathematics, the synthesis of morphisms and of transformations is so intimate that one can speak of a veritable fusion. […] Essentially, categories are sets of morphisms organized into operatory systems.
- 2007 November, Steven Dale Cutkosky, Toroidalization of Dominant Morphisms of 3-Folds, Memoirs of the American Mathematical Society, Volume 190, Number 890, page 3,
- The proof of toroidalization of morphisms of 3-folds to surfaces in [C3] breaks up into two parts: a reduction to prepared morphisms and then a proof of toroidalization of prepared morphisms from n-folds to [surfaces] in [CK].
- 1982, Israel Program for Scientific Translations (translator), Lev J. Leifman (editor of translation), N. N. ?encov, Statistical Decision Rules and Optimal Inference, American Mathematical Society, Translations of Mathematical Monographs, Volume 53, page 50,
- (biology) Being or having distinct variants of a plant or animal species in the same locale; polymorphism.
Synonyms
- (category theory): arrow, map
Derived terms
- bimorphism
- epimorphism
- monomorphism
Related terms
- automorphism
- diffeomorphism
- homeomorphism
- homomorphism
- isomorphism
Translations
See also
- bijection
- function
- functor
- isometry
- measurable function
morphism From the web:
- what is morphism in biology
- what does dimorphism mean
- what is morphism in mathematics
- polymorphism
- glassmorphism
- metamorphism
- isomorphism
- what is body morphism
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