Preimagen

Preimagen, or preimage in English, is a concept in mathematics referring to the set of elements that map to a given subset under a function. If f: X → Y and B ⊆ Y, the preimage of B is f^{-1}(B) = { x ∈ X | f(x) ∈ B }. The term is not the inverse function unless f is bijective; in general, f^{-1} denotes the preimage operation, not an inverse map.

Basic properties include monotonicity and compatibility with set operations: if A ⊆ B ⊆ Y, then f^{-1}(A) ⊆ f^{-1}(B). Preimages preserve unions and intersections: f^{-1}(A ∪ B) = f^{-1}(A) ∪ f^{-1}(B) and f^{-1}(A ∩ B) = f^{-1}(A) ∩ f^{-1}(B). Also f^{-1}(Y \ A) = X \ f^{-1}(A). For a single point y ∈ Y, f^{-1}({y}) is called the fiber over y.

In topology, preimages underpin continuity: a function f: X → Y between topological spaces is continuous exactly when the preimage of every open set in Y is open in X. In measure and probability theory, if f is measurable, the preimage of a measurable set is measurable, enabling the definition of pullbacks of measures and the expression of events in the domain as f^{-1}(B).

The concept applies broadly across set theory, topology, and analysis, and is fundamental for solving equations, inverse problems, and in describing how subsets of the codomain correspond to subsets of the domain.