More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) f ". Similarly, the inverse image (or preimage) of a given subset B of the codomain of f, is the set of all elements of the domain that map to the members of B.
Image and inverse image may also be defined for general binary relations, not just functions.
The traditional notations used in the previous section can be confusing. An alternative is to give explicit names for the image and preimage as functions between power sets:
With respect to the algebra of subsets described above, the inverse image function is a lattice homomorphism, while the image function is only a semilattice homomorphism (i.e., it does not always preserve intersections).