different between preimage vs injective

English

Etymology

pre- +? image.

Pronunciation

• Rhymes: -?m?d?

Noun

preimage (plural preimages)

1. (mathematics) For a given function, the set of all elements of the domain that are mapped into a given subset of the codomain; (formally) given a function ƒ : X ? Y and a subset B ? Y, the set ƒ^?1(B) = {x ? X : ƒ(x) ? B}.
   • 1967 [Academic Press], Francois Treves, Topological Vector Spaces, Distributions and Kernels, 2006, Dover, page 22,

     The preimage of a neighborhood U of 0 in E must be a neighborhood of (0,x), since (0,x) is mapped into 0.

   • 2003, Sergei K. Lando, Alexander K. Zvonkin, Graphs on Surfaces and Their Applications, Springer, page 56,

     Previously, maps and hypermaps were constructed as the preimages of a segment joining two of the three critical values.

   • 2005, Oded Goldreich, Foundations of Cryptography: A Primer, now Publishers, page 24,

     Loosely speaking, saying that a function f is one-way implies that given y (in the range of f) it is infeasible to find a preimage of y under f.

Synonyms

• (set of all elements that map into a given subset of the codomain of a function): inverse image

Hyponyms

• (set of all elements that map into a given subset of the codomain of a function): kernel

Derived terms

• preimage attack

Related terms

• image

Translations

Further reading

• Image (mathematics) on Wikipedia.Wikipedia
• Kernel (linear algebra) on Wikipedia.Wikipedia

English

Etymology

(This etymology is missing or incomplete. Please add to it, or discuss it at the Etymology scriptorium. Particularly: “cf. discussion page”)This term was introduced by Nicolas Bourbaki in his treatise Éléments de mathématique.

Pronunciation

• IPA^(key): /?n?d??kt?v/
• Rhymes: -?kt?v

Adjective

injective (not comparable)

1. (mathematics) of, relating to, or being an injection: such that each element of the image (or range) is associated with at most one element of the preimage (or domain); inverse-deterministic

   Synonym: one-to-one

Derived terms

Related terms

• injectively

Translations

French

Pronunciation

• IPA^(key): /??.??k.tiv/
• Homophone: injectives

Adjective

injective

1. feminine singular of injectif