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)