Image (mathematics)

Last updated
f is a function from domain X to codomain Y. The yellow oval inside Y is the image of f. Codomain2.SVG
f is a function from domain X to codomain Y. The yellow oval inside Y is the image of f.

In mathematics, the image of a function is the set of all output values it may produce.

Contents

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 (or through) f ". Similarly, 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 : XY is a function from the set X to the set Y.

Image of an element

If x is a member of X, then the image of x under f, denoted f(x), [1] is the value of f when applied to x.f(x) is alternatively known as the output of f for argument x.

Image of a subset

The image of a subset AX under f, denoted , is the subset of Y which can be defined using set-builder notation as follows: [2] [3]

When there is no risk of confusion, is simply written as . This convention is a common one; the intended meaning must be inferred from the context. This makes f[.] a function whose domain is the power set of X (the set of all subsets of X), and whose codomain is the power set of Y. See § Notation below for more.

Image of a function

The image of a function is the image of its entire domain, also known as the range of the function. [4] This usage should be avoided because the word "range" is also commonly used to mean the codomain of f.

Generalization to binary relations

If R is an arbitrary binary relation on X × Y, then the set { y∈Y | xRy for some xX } is called the image, or the range, of R. Dually, the set { xX | xRy for some y∈Y } 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 BY under f, denoted by , is the subset of X defined by

Other notations include f −1 (B) [5] and f  (B). [6] The inverse image of a singleton, denoted by f −1[{y}] or by f −1[y], is also called the fiber or fibre over y or the level set of y. The set of all the fibers over the elements of Y is a family of sets indexed by Y.

For example, for the function f(x) = x2, the inverse image of {4} would be {−2, 2}. Again, if there is no risk of confusion, f −1[B] can be denoted by f −1(B), and f −1 can also be thought of 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, although it coincides with the usual one for bijections in that the inverse image of B under f is the image of B under f −1.

Notation for image and inverse image

The traditional notations used in the previous section can be confusing. An alternative [7] is to give explicit names for the image and preimage as functions between power sets:

Arrow notation

Star notation

Other terminology

Examples

  1. f: {1, 2, 3} → {a, b, c, d} defined by
    The image of the set {2, 3} under f is f({2, 3}) = {a, c}. The image of the function f is {a, c}. The preimage of a is f −1({a}) = {1, 2}. The preimage of {a, b} is also {1, 2}. The preimage of {b, d} is the empty set {}.
  2. f: RR defined by f(x) = x2.
    The image of {−2, 3} under f is f({−2, 3}) = {4, 9}, and the image of f is R+ (the set of all positive real numbers and zero). The preimage of {4, 9} under f is f −1({4, 9}) = {−3, −2, 2, 3}. The preimage of set N = {nR | n < 0} under f is the empty set, because the negative numbers do not have square roots in the set of reals.
  3. f: R2R defined by f(x, y) = x2 + y2.
    The fibre f −1({a}) 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 a > 0, then the fiber f −1({a}) is the set of all (x, y) ∈ R2 satisfying the equation of the origin-concentric ring x2 + y2 = a.)
  4. If M is a manifold and π: TMM is the canonical projection from the tangent bundle TM to M, then the fibres of π are the tangent spaces Tx(M) for xM. This is also an example of a fiber bundle.
  5. A quotient group is a homomorphic image.

Properties

Counter-examples based on the real numbers
defined by
showing that equality generally need
not hold for some laws:
Image showing non-equal sets:
f
(
A
1
[?]
A
2
)
[?]
f
(
A
1
)
[?]
f
(
A
2
)
.
{\displaystyle f\left(A_{1}\cap A_{2}\right)\subsetneq f\left(A_{1}\right)\cap f\left(A_{2}\right).}
The sets
A
1
=
[
-
4
,
2
]
{\displaystyle A_{1}=[-4,2]}
and
A
2
=
[
-
2
,
4
]
{\displaystyle A_{2}=[-2,4]}
are shown in blue immediately below the
x
{\displaystyle x}
-axis while their intersection
A
3
=
[
-
2
,
2
]
{\displaystyle A_{3}=[-2,2]}
is shown in green. Image preimage conterexample intersection.gif
Image showing non-equal sets: The sets and are shown in blue immediately below the -axis while their intersection is shown in green.
f
(
f
-
1
(
B
3
)
)
[?]
B
3
.
{\displaystyle f\left(f^{-1}\left(B_{3}\right)\right)\subsetneq B_{3}.} Image preimage conterexample bf.gif
f
-
1
(
f
(
A
4
)
)
[?]
A
4
.
{\displaystyle f^{-1}\left(f\left(A_{4}\right)\right)\supsetneq A_{4}.} Image preimage conterexample fb.gif

General

For every function and all subsets and the following properties hold:

ImagePreimage

(equal if , e.g. is surjective) [10] [11]

(equal if is injective) [10] [11]
[10]
[12] [12]
[12] [12]

Also:

Multiple functions

For functions and with subsets and the following properties hold:

Multiple subsets of domain or codomain

For function and subsets and the following properties hold:

ImagePreimage
[12] [13]
[12] [13]
(equal if is injective [14] )
[12]
(equal if is injective [14] )
[12]

(equal if is injective)

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, S can be infinite, even uncountably infinite.)

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 (i.e., it does not always preserve intersections).

See also

Notes

  1. "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-08-28.
  2. "5.4: Onto Functions and Images/Preimages of Sets". Mathematics LibreTexts. 2019-11-05. Retrieved 2020-08-28.
  3. Paul R. Halmos (1968). Naive Set Theory. Princeton: Nostrand. Here: Sect.8
  4. Weisstein, Eric W. "Image". mathworld.wolfram.com. Retrieved 2020-08-28.
  5. "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-08-28.
  6. Dolecki & Mynard 2016, pp. 4-5.
  7. Blyth 2005, p. 5.
  8. Jean E. Rubin (1967). Set Theory for the Mathematician . Holden-Day. p. xix. ASIN   B0006BQH7S.
  9. M. Randall Holmes: Inhomogeneity of the urelements in the usual models of NFU, December 29, 2005, on: Semantic Scholar, p. 2
  10. 1 2 3 See Halmos 1960 , p. 39
  11. 1 2 See Munkres 2000 , p. 19
  12. 1 2 3 4 5 6 7 8 See p.388 of Lee, John M. (2010). Introduction to Topological Manifolds, 2nd Ed.
  13. 1 2 Kelley 1985 , p.  85
  14. 1 2 See Munkres 2000 , p. 21

Related Research Articles

In mathematics, a partial functionf from a set X to a set Y is a function from a subset S of X to Y. The subset S, that is, the domain of f viewed as a function, is called the domain of definition of f. If S equals X, that is, if f is defined on every element in X, then f is said to be total.

Surjective function Function such that every element has a preimage (mathematics)

In mathematics, a surjective function is a function f that maps something to every element y; that is, for every y, there is an x such that f(x) = y. In other words, every element of the function's codomain is the image of at least one element of its domain. It is not required that x be unique; the function f may map one or more elements of X to the same element of Y.

In mathematical analysis and in probability theory, a σ-algebra on a set X is a collection Σ of subsets of X that includes X itself, is closed under complement, and is closed under countable unions.

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. This is in direct analogy to the definition that a continuous function between topological spaces preserves the topological structure: the preimage of any open set is open. In real analysis, measurable functions are used in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability space is known as a random variable.

Injective function Function that preserves distinctness

In mathematics, an injective function is a function f that maps distinct elements to distinct elements; that is, f(x1) = f(x2) implies x1 = x2. In other words, every element of the function's codomain is the image of at most one element of its domain. The term one-to-one function must not be confused with one-to-one correspondence that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain.

General topology Branch of topology

In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.

Function (mathematics) Mapping that associates a single output value to each input

In mathematics, a function is a binary relation between two sets that associates each element of the first set to exactly one element of the second set. Typical examples are functions from integers to integers, or from the real numbers to real numbers.

Indicator function Mathematical function

In mathematics, an indicator function or a characteristic function of a subset A of a set X is a function defined from X to the two elements set {0, 1}, typically denoted as , and it indicates whether an element in X belongs to A; if an element in X belongs to A, and if does not belong to A. It is also denoted by to emphasize the fact that this function identifies the subset A of X.

Symmetric difference Subset of the elements that belong to exactly one among two sets

In mathematics, the symmetric difference of two sets, also known as the disjunctive union, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets and is .

In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition. They were first formalized by Kazimierz Kuratowski, and the idea was further studied by mathematicians such as Wacław Sierpiński and António Monteiro, among others.

In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function is open if for any open set in the image is open in Likewise, a closed map is a function that maps closed sets to closed sets. A map may be open, closed, both, or neither; in particular, an open map need not be closed and vice versa.

Bijection, injection and surjection Properties of mathematical functions

In mathematics, injections, surjections, and bijections are classes of functions distinguished by the manner in which arguments and images are related or mapped to each other.

A Dynkin system, named after Eugene Dynkin, is a collection of subsets of another universal set satisfying a set of axioms weaker than those of 𝜎-algebra. Dynkin systems are sometimes referred to as 𝜆-systems or d-system. These set families have applications in measure theory and probability.

In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations that are defined on it.

In mathematics, the term fiber or fibre can have two meanings, depending on the context:

  1. In naive set theory, the fiber of the element y in the set Y under a map f : XY is the inverse image of the singleton under f.
  2. In algebraic geometry, the notion of a fiber of a morphism of schemes must be defined more carefully because, in general, not every point is closed.

In mathematics, a nonempty collection of sets is called a σ-ring if it is closed under countable union and relative complementation.

In mathematics, a content is a set function that is like a measure, but a content must only be finitely additive, whereas a measure must be countably additive. A content is a real function defined on a collection of subsets such that

In topology, the pasting or gluing lemma, and sometimes the gluing rule, is an important result which says that two continuous functions can be "glued together" to create another continuous function. The lemma is implicit in the use of piecewise functions. For example, in the book Topology and Groupoids, where the condition given for the statement below is that and .

In mathematics, a polyadic space is a topological space that is the image under a continuous function of a topological power of an Alexandroff one-point compactification of a discrete space.

References

This article incorporates material from Fibre on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.