preimagee

Preimagee is commonly a misspelling of the mathematical term preimage. In standard usage, the preimage refers to the inverse image of a subset under a function. The term preimage is preferred in formal writing, and readers encountering preimagee should recognize it as an orthographic error rather than a distinct concept.

Formally, let f be a function from a set X to a set Y, and let S be a subset of Y. The preimage of S under f is the set f^{-1}(S) = { x in X | f(x) ∈ S }. If S consists of a single element {y}, the preimage is sometimes called the fiber over y and denoted f^{-1}({y}). The preimage operation is defined for any subset S of Y, not just single points.

Key properties include: f^{-1}(∅) = ∅ and f^{-1}(Y) = X; f^{-1}(A ∪ B) = f^{-1}(A) ∪ f^{-1}(B); f^{-1}(A ∩ B) = f^{-1}(A) ∩ f^{-1}(B). If f is a continuous function between topological spaces, the preimage of an open set is open; more generally, continuity can be characterized by the behavior of preimages with respect to open sets.

Examples illustrate the concept: with f(x) = x^2, f^{-1}({1}) = {−1, 1}, and f^{-1}((2, 3)) = (−√3, −√2) ∪ (√2, √3). In probability, the preimage of an event B under a random variable is the set of outcomes mapping into B.

Notes: the term preimagee appears as a nonstandard spelling. In technical writing, use preimage to avoid confusion. See also image, inverse image, fiber, and pullback.