PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (210)
Orphanage
Unclass'd (1)
Unproven (472)
Corrections (37)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Low Entry average rating: Very low
inverse image (Definition)

Let $ f: A \longrightarrow B$ be a function, and let $ U \subset B$ be a subset. The inverse image of $ U$ is the set $ f^{-1}(U) \subset A$ consisting of all elements $ a \in A$ such that $ f(a) \in U$.

The inverse image commutes with all set operations: For any collection $ \{U_i\}_{i \in I}$ of subsets of $ B$, we have the following identities for

  1. Unions:
    $\displaystyle f^{-1}\left(\bigcup_{i \in I} U_i\right) = \bigcup_{i \in I} f^{-1}(U_i)$
  2. Intersections:
    $\displaystyle f^{-1}\left(\bigcap_{i \in I} U_i\right) = \bigcap_{i \in I} f^{-1}(U_i)$
and for any subsets $ U$ and $ V$ of $ B$, we have identities for
  1. Complements:
    $\displaystyle \left(f^{-1}(U)\right)^\complement = f^{-1}(U^\complement)$
  2. Set differences:
    $\displaystyle f^{-1}(U \setminus V) = f^{-1}(U) \setminus f^{-1}(V)$
  3. Symmetric differences:
    $\displaystyle f^{-1}(U \bigtriangleup V) = f^{-1}(U) \bigtriangleup f^{-1}(V)$
In addition, for $ X \subset A$ and $ Y \subset B$, the inverse image satisfies the miscellaneous identities
  1. $ (f\vert _X)^{-1}(Y)=X\cap f^{-1}(Y)$
  2. $ f\left(f^{-1}(Y)\right) = Y\cap f(A)$
  3. $ X \subset f^{-1}(f(X))$, with equality if $ f$ is injective.



"inverse image" is owned by djao. [ full author list (2) | owner history (1) ]
(view preamble)

View style:

See Also: mapping, direct image

Other names:  preimage

Attachments:
the inverse image commutes with set operations (Proof) by matte
Log in to rate this entry.
(view current ratings)

Cross-references: injective, equality, satisfies, symmetric differences, set differences, complements, intersections, unions, identities, collection, operations, subset, function
There are 44 references to this entry.

This is version 5 of inverse image, born on 2001-10-21, modified 2003-07-30.
Object id is 442, canonical name is InverseImage.
Accessed 12412 times total.

Classification:
AMS MSC: 03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )
Pending Errata and Addenda
None.
[ View all 2 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)