Inverses and Composites

Abstract

Associated with every function f, from X to Y, say, there is a function from $\wp (X)$ to $\wp (Y)$, namely the function (frequently called f also) that assigns to each subset A of X the image subset f(A) of Y. The algebraic behavior of the mapping A → f(A) leaves something to be desired. It is true that if {A _i } is a family of subsets of X, then $f\left( {\bigcup {_i A_i } } \right) = \bigcup {_i f\left( {A_i } \right)}$(proof?), but the corresponding equation for intersections is false in general (example?), and the connection between images and complements is equally unsatisfactory.