præimages

A præimage, also known as a preimage or inverse image, is a concept in set theory and mathematics related to functions. Given a function f mapping a set A to a set B, and a subset S of the codomain B, the præimage of S under f, denoted f⁻¹(S), is the set of all elements in the domain A that map to elements within S. In simpler terms, it's the collection of inputs that produce outputs belonging to a specific set.

The definition can be expressed formally as: f⁻¹(S) = {x ∈ A | f(x) ∈ S}. It's important to distinguish the præimage of a set from the image of a set. The image of a set T ⊆ A, denoted f(T), is the set of all outputs obtained by applying f to the elements of T. The præimage is concerned with reversing the mapping, asking which elements in the domain correspond to a given subset of the codomain.

For instance, if f(x) = x² and S = {4, 9}, then the præimage of S under f is the set {-3, 3, -2, 2}. This is because f(-3) = 9, f(3) = 9, f(-2) = 4, and f(2) = 4, and no other real numbers squared result in 4 or 9. The præimage of a single element {y} from the codomain is sometimes referred to as the fiber over y. If f is a constant function, the præimage of any non-empty subset of its codomain will be the entire domain.