Here's what i originally read:
Given a map
:A->B, we can associate with any element b
B the set of those elements of A that take the value b under
. This set is the pre-image of b under
and is written
This set is given by
= {a
A:
(a)=b}.
The notation suggests that the inverse is a map with domain B but unless we are very lucky the codomain is not A. The above equation shows that the values of the inverse map are actually subsets of A so that the codomain is the powerset P(A).
More generally, we can make the following definition:
For all subsets Y of B,
(Y) = {a
A:
(a)
Y}
of the pre-image of any subset Y contained in B. This delivers a map
: P(B)->P(A)