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The preimage of an output value is the set of input values that produce . More generally, evaluating f {\displaystyle f} at each element of a given subset A {\displaystyle A} of its domain X {\displaystyle X} produces a set, called the " image of A {\displaystyle A} under (or through) f {\displaystyle f} ".
Moreover, f is the composition of the canonical projection from f to the quotient set, and the bijection between the quotient set and the codomain of . The composition of two surjections is again a surjection, but if g ∘ f {\displaystyle g\circ f} is surjective, then it can only be concluded that g {\displaystyle g} is surjective (see figure).
In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable.
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] → B 0. Because of this example, in a general category the pullback of a morphism f and a monomorphism g can be thought of as the "preimage" under ...
The fibers of are that line and all the straight lines parallel to it, which form a partition of the plane . More generally, if f {\displaystyle f} is a linear map from some linear vector space X {\displaystyle X} to some other linear space Y {\displaystyle Y} , the fibers of f {\displaystyle f} are affine subspaces of X {\displaystyle X ...
If f : X → Y is surjective and B is a subset of Y, then f(f −1 (B)) = B. Thus, B can be recovered from its preimage f −1 (B). For example, in the first illustration in the gallery, there is some function g such that g(C) = 4. There is also some function f such that f(4) = C.
Form 3 is an SEC filing filed with the US Securities and Exchange Commission to indicate a preliminary insider transaction by an officer, director, or beneficial (10%) owner of the company's securities. These are typically seen after a company IPOs when insiders make their first transactions.
In mathematics, particularly in the field of differential topology, the preimage theorem is a variation of the implicit function theorem concerning the preimage of particular points in a manifold under the action of a smooth map.