# preimage

## English

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pre- +‎ image.

### 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}.
The preimage of ${\displaystyle \{4,9\}}$ under the function ${\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} :f(x)=x^{2}}$ is the set ${\displaystyle \{-3,-2,+2,+3\}}$.
• 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