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Similarly, the inverse image (or preimage) of a given subset of the codomain is the set of all elements of that map to a member of . The image of the function f {\displaystyle f} is the set of all output values it may produce, that is, the image of X {\displaystyle X} .
If and are the domain and image of , respectively, then the fibers of are the sets in {():} = {{: =}:}which is a partition of the domain set .Note that must be restricted to the image set of , since otherwise () would be the empty set which is not allowed in a partition.
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
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.
Some authors call a function : between two topological spaces proper if the preimage of every compact set in is compact in . Other authors call a map f {\displaystyle f} proper if it is continuous and closed with compact fibers ; that is if it is a continuous closed map and the preimage of every point in Y {\displaystyle Y} is compact .
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.