different between bijection vs isomorphism

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

English

Etymology

iso- +? -morphism

Pronunciation

Noun

isomorphism (plural isomorphisms)

1. Similarity of form
   • 1984 Brigitte Asbach-Schnitker, "Introduction", Mercury or The Secret and Swift Messenger, ?ISBN.

     The postulated isomorphism between words and things constitutes the characterizing feature of all philosophically based universal languages.

   1. (biology) the similarity in form of organisms, which may be due to convergent evolution or shared genetic background, e.g. an algae species in which the haploid and diploid life stages are indistinguishable based on morphology.
   2. (chemistry) the similarity in the crystal structures of similar chemical compounds
      • 1874 C. Rammelsberg, "Crystallographic and chemical relations of the natural sulphides, arsenides, and sulpharsenides", The Chemical News and Journal of Physical Science, page 197.

        The isomorphism of compounds does not prove the isomorphism of their respective constituents.

   3. (sociology) the similarity in the structure or processes of different organizations
2. A one-to-one correspondence
   1. (group algebra) A bijection f such that both f and its inverse f ^?1 are homomorphisms, that is, structure-preserving mappings.
   2. (computer science) a one-to-one correspondence between all the elements of two sets, e.g. the instances of two classes, or the records in two datasets
   3. (category theory) A morphism which has an inverse; the composition of the morphism and its inverse yields either one of two identity morphisms (depending on the order of composition).

Synonyms

• (in category theory): iso

Antonyms

• anisomorphism

Related terms

• isomorphic
• isomorphous

Translations

See also

• Isomorphism on Wikipedia.Wikipedia
• Isomorphism on Encyclopedia of Mathematics
• isomorphism on nLab
• Isomorphism on Wolfram MathWorld
• Graph Isomorphism on Wolfram MathWorld
• Natural Isomorphism on Wolfram MathWorld