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)=x2 is {-1,1}, because (-1)2=1 and 12=1, and for no other x is this true.
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