If you have a map f:A->B , i've read that its domain and codomain are the powersets of B and A respectively. Why is this so? Why aren't they simply B and A?
yes, they are still the same
a function $\displaystyle f:A \mapsto B$ has domain A and codomain B, since the domain is the set of inputs and the codomain is the set of outputs. Now an inverse function is a function that does pretty much the reverse of f. as in it takes the output from f and brings it back to the original input. when you consider that, you will realize that it takes the output from f as its input and the input from f as its output, and therefore its domain would be B and it's codomain A. note that an invertible function will be bijective
What Jhevon gave you is the standard and usual definition.
However, some set theory texts have introduced two new types of functions.
Given a function $\displaystyle f:A \mapsto B$ there are two new functions $\displaystyle \overleftarrow f :P(B) \mapsto P(A)$ and $\displaystyle \overrightarrow f :P(A) \mapsto P(B)$
Could this be what you have in mind?
The definitions are rather difficult to type even in TeX.
If this is it, then I will post a pdf file containing the definitions and examples.
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)
What your text has given exactly what I posted above and used the symbol $\displaystyle \overleftarrow f $ instead of $\displaystyle f^{ - 1} $. Many of us object to that last notation because it is easily confused confused with the reciprocal. Above you said that that text said $\displaystyle B^A$ was not no longer is wide use. Well I know that the text is mistaken with that judgment. Do you have any idea why $\displaystyle B^A$ was chosen (probably by Cantor) for the set of the set of all mappings from A to B? Well, if A has n elements and B has m elements then there are $\displaystyle m^n$ mappings from A to B. And that idea works well in transfinite arithmetic.