Let 
be a map between sets 
and 
.
Let 
. Then the preimage of 
under 
is denoted by 
, and is the set of all elements of 
that map to elements in 
under 
. Thus
 |
(1)
|
One is not to be mislead by the notation into thinking of the preimage as having to do with an inverse of 
.
The preimage is defined whether 
has an inverse or not. Note however that if 
does have an inverse, then the preimage 
is exactly the image of
under the inverse map, thus justifying
the perhaps slightly misleading notation.
For any 
,
it is true that
 |
(2)
|
with equality occurring, if 
is surjective, and for any subset 
, it is true that
 |
(3)
|
with equality occurring if 
is injective.
Preimages occur in a variety of subjects, the most persistent of these being topology, where a map is continuous, by definition, if the preimage of every open set is open.
