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The image of the function is the set of all output values it may produce, that is, the image of . The preimage of f {\displaystyle f} , that is, the preimage of Y {\displaystyle Y} under f {\displaystyle f} , always equals X {\displaystyle X} (the domain of f {\displaystyle f} ); therefore, the former notion is rarely used.
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.
This function maps each image to its unique preimage. The composition of two bijections is again a bijection, but if g ∘ f {\displaystyle g\circ f} is a bijection, then it can only be concluded that f {\displaystyle f} is injective and g {\displaystyle g} is surjective (see the figure at right and the remarks above regarding injections and ...
On the other hand, the inverse image or preimage under f of an element y of the codomain Y is the set of all elements of the domain X whose images under f equal y. [6] In symbols, the preimage of y is denoted by f − 1 ( y ) {\displaystyle f^{-1}(y)} and is given by the equation
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.
Also called an injection or, sometimes, one-to-one function. In other words, every element of the function's codomain is the image of at most one element of its domain. Surjective function: has a preimage for every element of the codomain, that is, the codomain equals the image.
In mathematics, specifically in algebraic topology and algebraic geometry, an inverse image functor is a contravariant construction of sheaves; here “contravariant” in the sense given a map :, the inverse image functor is a functor from the category of sheaves on Y to the category of sheaves on X.
The first isomorphism theorem for monoids states that this quotient monoid is naturally isomorphic to the image of f (which is a submonoid of N; for the congruence relation). This is very different in flavour from the above examples. In particular, the preimage of the identity element of N is not enough to determine the kernel of f.