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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the image of a set under a given function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 is the set of all possible function outputs when taking each element of the set, successively, as the function's argument.
Definition
If f : X ? Y is a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 from set X to set Y and x is a member of X, then f(x), the image of x under f, is a unique member of Y that f associates with x.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, the image of a set under a given function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 is the set of all possible function outputs when taking each element of the set, successively, as the function's argument.

Definition


If f : X ? Y is a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 from set X to set Y and x is a member of X, then f(x), the image of x under f, is a unique member of Y that f associates with x. The image under f of the entire domain X is often called the range of f, and is a subset of the codomain
Codomain

In mathematics, the codomain, range or target set, of a function , described symbolically as ' : ' ? ', is the set ' into which all of the output of the function is constrained to fall....
 Y.

The image of a subset
Subset

In mathematics, especially in set theory, a Set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide....
 A ? X under f is the subset of Y defined by

f[A] = .


When there is no risk of confusion, f[A] is simply written as f(A). An alternative notation for f[A] that is common in the older literature mathematical logic
Mathematical logic

Mathematical logic is a subfield of mathematics and logic with close connections to computer science and philosophical logic. The field includes the mathematical study of logic and the applications of formal logic to other areas of mathematics....
 and still preferred by some set theorists
Set theory

Set theory is the branch of mathematics that studies Set , which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics....
, is f "A.

Given this definition, the image of f becomes a function whose domain is the power set
Power set

In mathematics, given a Set S, the power set of S, written , P, ℘ or Power set#Representing subsets as functions, is the set of all subsets of S....
 of X (the set of all subset
Subset

In mathematics, especially in set theory, a Set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide....
s of X), and whose codomain
Codomain

In mathematics, the codomain, range or target set, of a function , described symbolically as ' : ' ? ', is the set ' into which all of the output of the function is constrained to fall....
 is the power set of Y. The same notation can denote either the function f or its image. This convention is a common one; the intended meaning must be inferred from the context.

The preimage or inverse image of a set B ? Y under f is the subset of X defined by

f −1[B] = .


The inverse image of a singleton
Singleton (mathematics)

In mathematics, a singleton is a Set with unique element. For example, the set is a singleton....
, f −1[], is a fiber
Fiber (mathematics)

In mathematics, the fiber of a point y under a function f : X ? Y is the inverse relation of under f, that is, ...
 (also spelled fibre) or a level set
Level set

In mathematics, a level set of a real number-valued function f of n variables is a set of the formwhere c is a constant. That is, it is the set where the function takes on a given constant value....
.

Again, if there is no risk of confusion, we may denote f −1[B] by f −1(B), and think of f −1 as a function from the power set of Y to the power set of X. The notation f −1 should not be confused with that for inverse function
Inverse function

In mathematics, if ƒ is a function from A to B then an inverse function for ƒ is a function in the opposite direction, from B to A, with the property that a round trip from A to B to A returns each element of the initial set to itself....
. The two coincide only if f is a bijection
Bijection

In mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property that, for every y in Y, there is exactly one x in X such that f = y....
.

f can also be seen as a family of sets indexed by Y, which leads to the notion of a fibred category
Fibred category

Fibred categories are abstract entities in mathematics used to provide a general framework for descent theory. They formalise the various situations in geometry and algebra in which inverse images of objects such as vector bundles can be defined....
.

Uniform arrow notations

The traditional notations used in the previous section can be confusing. An alternative is to explicitly write the image and preimage as two functions in their own right: with and with . If we consider the powerset as a poset ordered by inclusion
Subset

In mathematics, especially in set theory, a Set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide....
, then the image and preimage functions are monotone
Monotonic function

In mathematics, a monotonic function is a function which preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....
.

Examples