inverse image
Let be a function, and let be a subset. The inverse image of is the set consisting of all elements such that .
The inverse image commutes with all set operations: For any collection
of subsets of , we have the following identities
for
- 1.
Unions:
- 2.
Intersections
:
and for any subsets and of , we have identities for
- 3.
Complements
:
- 4.
Set differences
:
- 5.
Symmetric differences
:
In addition, for and , the inverse image satisfies the miscellaneous identities
- 6.
- 7.
- 8.
, with equality if is injective
.
Title | inverse image |
Canonical name | InverseImage |
Date of creation | 2013-03-22 11:51:58 |
Last modified on | 2013-03-22 11:51:58 |
Owner | djao (24) |
Last modified by | djao (24) |
Numerical id | 10 |
Author | djao (24) |
Entry type | Definition |
Classification | msc 03E20 |
Classification | msc 46L05 |
Classification | msc 82-00 |
Classification | msc 83-00 |
Classification | msc 81-00 |
Synonym | preimage |
Related topic | Mapping |
Related topic | DirectImage |