, the image of an input value
is the single output value produced by
The preimage of an output value
is the set of input values that produce
produces a set, called the "image of
Similarly, the inverse image (or preimage) of a given subset
is the set of all output values it may produce, that is, the image of
Image and inverse image may also be defined for general binary relations, not just functions.
The word "image" is used in three related ways.
in the function's domain such that
Using set-builder notation, this definition can be written as[1][2]
The image of a function is the image of its entire domain, also known as the range of the function.
[3] This last usage should be avoided because the word "range" is also commonly used to mean the codomain of
is an arbitrary binary relation on
is called the image, or the range, of
The preimage or inverse image of a set
[4] The inverse image of a singleton set, denoted by
The set of all the fibers over the elements of
is a family of sets indexed by
can also be thought of as a function from the power set of
should not be confused with that for inverse function, although it coincides with the usual one for bijections in that the inverse image of
The traditional notations used in the previous section do not distinguish the original function
; likewise they do not distinguish the inverse function (assuming one exists) from the inverse image function (which again relates the powersets).
Given the right context, this keeps the notation light and usually does not cause confusion.
But if needed, an alternative[5] is to give explicit names for the image and preimage as functions between power sets: For every function
the following properties hold: Also: For functions
the following properties hold: For function
the following properties hold: The results relating images and preimages to the (Boolean) algebra of intersection and union work for any collection of subsets, not just for pairs of subsets: (Here,
With respect to the algebra of subsets described above, the inverse image function is a lattice homomorphism, while the image function is only a semilattice homomorphism (that is, it does not always preserve intersections).
This article incorporates material from Fibre on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.