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For the function that maps a Person to their Favorite Food, the image of Gabriela is Apple. The preimage of Apple is the set {Gabriela, Maryam}. The preimage of Fish is the empty set. The image of the subset {Richard, Maryam} is {Rice, Apple}. The preimage of {Rice, Apple} is {Gabriela, Richard, Maryam}.
A function : is monotone in this topological sense if and only if it is non-increasing or non-decreasing, which is the usual meaning of "monotone function" in real analysis. A function between topological spaces is (sometimes) called a proper map if every fiber is a compact subspace of its domain. However, many authors use other non-equivalent ...
Measurable function: the preimage of each measurable set is measurable. Borel function: the preimage of each Borel set is a Borel set. Baire function called also Baire measurable function: obtained from continuous functions by transfinite iteration of the operation of forming pointwise limits of sequences of functions.
The function f : R → R defined by f(x) = x 3 − 3x is surjective, because the pre-image of any real number y is the solution set of the cubic polynomial equation x 3 − 3x − y = 0, and every cubic polynomial with real coefficients has at least one real root.
Let : be any function. If is any set then its preimage := under is necessarily an -saturated set.In particular, every fiber of a map is an -saturated set.. The empty set = and the domain = are always saturated.
Let V and W be vector spaces over a field (or more generally, modules over a ring) and let T be a linear map from V to W.If 0 W is the zero vector of W, then the kernel of T is the preimage of the zero subspace {0 W}; that is, the subset of V consisting of all those elements of V that are mapped by T to the element 0 W.
Preimages of sets under functions can be described as pullbacks as follows: Suppose f : A → B, B 0 ⊆ B. Let g be the inclusion map B 0 ↪ B. Then a pullback of f and g (in Set) is given by the preimage f −1 [B 0] together with the inclusion of the preimage in A. f −1 [B 0] ↪ A. and the restriction of f to f −1 [B 0] f −1 [B 0 ...
In a category with all finite limits and colimits, the image is defined as the equalizer (,) of the so-called cokernel pair (,,), which is the cocartesian of a morphism with itself over its domain, which will result in a pair of morphisms ,:, on which the equalizer is taken, i.e. the first of the following diagrams is cocartesian, and the second equalizing.