Definition: Let *f* :*X*->*Y* be a map and *A* a subset of *Y*. Then the *preimage* of *A* under *f*, denoted *f* ^{-1}(*A*), is equal to {*x* in *X* | *f* (*x*) in *A* }.

That is, the preimage of a set *A* under a map *f* is the set of elements of the domain of *f* that get mapped to an element of *A*. For example, the preimage of {1} under *f* :R->R defined by *f* (*x*)=*x*^{2} is {-1,1}, because (-1)^{2}=1 and 1^{2}=1, and for no other *x* is this true.