Image (mathematics)
In mathematics, the image of a function is the set of all output values it may produce.
More generally, evaluating a given function f at each element of a given subset A of its domain produces a set called the "image of A under f ". The inverse image or preimage of a given subset B of the codomain of f is the set of all elements of the domain that map to the members of B.
Image and inverse image may also be defined for general binary relations, not just functions.
Definition
The word "image" is used in three related ways. In these definitions, f : X → Y is a function from the set X to the set Y.Image of an element
If x is a member of X, then f = y is the image of x under f. y is alternatively known as the output of f for argument x.Image of a subset
The image of a subset A ⊆ X under f is the subset fWhen there is no risk of confusion, f
Image of a function
The image of a function is the image of its entire domain.Generalization to binary relations
If R is an arbitrary binary relation on X×Y, the set is called the image, or the range, of R. Dually, the set is called the domain of R.Inverse image
Let f be a function from X to Y. The preimage or inverse image of a set B ⊆ Y under f is the subset of X defined byThe inverse image of a singleton, denoted by f^{ −1}
For example, for the function f = x^{2}, the inverse image of would be. Again, if there is no risk of confusion, denote f^{ −1}
Notation for image and inverse image
The traditional notations used in the previous section can be confusing. An alternative is to give explicit names for the image and preimage as functions between powersets:Arrow notation
- with
- with
Star notation
- instead of
- instead of
Other terminology
- An alternative notation for f
used in mathematical logic and set theory is f "A. - Some texts refer to the image of f as the range of f, but this usage should be avoided because the word "range" is also commonly used to mean the codomain of f.
Examples
- f: → defined by The image of the set under f is f =. The image of the function f is. The preimage of a is f^{ −1} =. The preimage of is also. The preimage of is the empty set.
- f: R → R defined by f = x^{2}. The image of under f is f =, and the image of f is R^{+}. The preimage of under f is f^{ −1} =. The preimage of set N = under f is the empty set, because the negative numbers do not have square roots in the set of reals.
- f: R^{2} → R defined by f = x^{2} + y^{2}. The fibres f^{ −1} are concentric circles about the origin, the origin itself, and the empty set, depending on whether a > 0, a = 0, or a < 0, respectively.
- If M is a manifold and π: TM → M is the canonical projection from the tangent bundle TM to M, then the fibres of π are the tangent spaces T_{x} for x∈M. This is also an example of a fiber bundle.
- A quotient group is a homomorphic image.
Properties
Counter-examples based on f:ℝ→ℝ, x↦x^{2}, showing that equality generally need not hold for some laws: |
General
For every function and all subsets and, the following properties hold:Image | Preimage |
Also:
Multiple functions
Multiple subsets of domain or codomain
Image | Preimage |
The results relating images and preimages to the algebra of intersection and union work for any collection of subsets, not just for pairs of subsets: