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Preimage vs Bijection - What's the difference?

preimage | bijection |

As nouns the difference between preimage and bijection

is that preimage is (mathematics) the set containing exactly every member of the domain of a function such that the member is mapped by the function onto an element of a given subset of the codomain of the function formally, of a subset b'' of the codomain ''y'' under a function ƒ, the subset of the domain ''x defined by while bijection is (set theory) a one-to-one correspondence, a function which is both a surjection and an injection.

preimage

English

(Function)

Noun

(en noun)
  • (mathematics) The set containing exactly every member of the domain of a function such that the member is mapped by the function onto an element of a given subset of the codomain of the function. Formally, of a subset B'' of the codomain ''Y'' under a function ƒ, the subset of the domain ''X defined by
    • f^{-1}(B) = \{x \in X : f(x) \in B\}.
    For example, the preimage of {4, 9} under the squaring function is the set {?3,?2,+2,+3}.

    Synonyms

    * inverse image

    Related terms

    * image

    bijection

    English

    (wikipedia bijection)

    Noun

    (en noun)
  • (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