Image (mathematics) - Inverse Image

Inverse Image

"Preimage" redirects here. For the cryptographic attack on hash functions, see preimage attack.

Let f be a function from X to Y. The preimage or inverse image of a set B ⊆ Y under f is the subset of X defined by

The inverse image of a singleton, denoted by f −1 or by f −1, is also called the fiber over y or the level set of y. The set of all the fibers over the elements of Y is a family of sets indexed by Y.

Again, if there is no risk of confusion, we may denote f −1 by f −1(B), and think of f −1 as a function from the power set of Y to the power set of X. The notation f −1 should not be confused with that for inverse function. The two coincide only if f is a bijection.