preimage

Preimage, in mathematics, refers to the set of all elements in the domain that map into a given set under a function. If f: X → Y is a function and A ⊆ Y, the preimage of A under f is f^{-1}(A) = { x ∈ X | f(x) ∈ A }. For a single element y ∈ Y, the preimage f^{-1}({y}) is called the fiber over y.

Notation and relation to image: The preimage is a subset of X, while the image of a subset B ⊆ X is f(B) ⊆ Y. Thus f^{-1}(A) generalizes taking inverse images of sets, not individual values. The preimage operator respects many set operations: f^{-1}(∅) = ∅, f^{-1}(Y) = X, f^{-1}(A ∪ B) = f^{-1}(A) ∪ f^{-1}(B), f^{-1}(A ∩ B) = f^{-1}(A) ∩ f^{-1}(B), and f^{-1}(Y \ A) = X \ f^{-1}(A).

Surjectivity and injectivity: For y ∈ Y, y ∈ Im(f) if and only if f^{-1}({y}) ≠ ∅. The function f is surjective onto Y precisely when f^{-1}({y}) ≠ ∅ for every y ∈ Y. If f is injective, then the preimage of each y in Im(f) is either empty or a singleton, and distinct y have disjoint preimages.

Applications: Preimages underpin definitions of continuity (preimage of open sets must be open), measure theory (pullback of sets), and various areas of algebra and topology. They provide a way to transfer structure from the codomain to the domain via inverse images.

Example: Let f: {a, b, c} → {1, 2} with f(a) = 1, f(b) = 1, f(c) = 2. Then f^{-1}({1}) = {a, b} and f^{-1}({2}) = {c}.