Originally Posted by

**Obstacle1** Here's what i originally read:

Given a map $\displaystyle \theta$:A->B, we can associate with any element b$\displaystyle \epsilon$B the set of those elements of A that take the value b under $\displaystyle \theta$. This set is the pre-image of b under $\displaystyle \theta$ and is written $\displaystyle \theta^{-1}$

This set is given by $\displaystyle \theta^{-1}$ = {a$\displaystyle \epsilon$A:$\displaystyle \theta$(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, $\displaystyle \theta^{-1}$(Y) = {a$\displaystyle \epsilon$A:$\displaystyle \theta$(a)$\displaystyle \epsilon$Y}

of the pre-image of any subset Y contained in B. This delivers a map

$\displaystyle \theta^{-1}$: P(B)->P(A)