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Target set of a mathematical function
In mathematics, a codomain or set of destination of a function is a set into which all of the outputs of the function are constrained to fall. It is the
Codomain
Association of one output to each input
set X is called the domain of the function and the set Y is called the codomain of the function. Functions were originally the idealization of how a varying
Function_(mathematics)
Mathematical function such that every output has at least one input
function's codomain, there exists at least one element x in the function's domain such that f(x) = y. In other words, for a function f : X → Y, the codomain Y
Surjective_function
Set of all things that may be the input of a mathematical function
function f : X → Y {\displaystyle f\colon X\to Y} , the set Y is called the codomain: the set to which all outputs must belong. The set of specific outputs
Domain_of_a_function
Function that preserves distinctness
that maps distinct elements of its domain to distinct elements of its codomain; that is, x1 ≠ x2 implies f(x1) ≠ f(x2) (equivalently by contraposition
Injective_function
Subset of a function's codomain
function may refer either to the codomain of the function, or the image of the function. In some cases the codomain and the image of a function are the
Range_of_a_function
Properties of mathematical functions
expressions from the codomain) are related or mapped to each other. A function maps elements from its domain to elements in its codomain. Given a function
Bijection, injection and surjection
Bijection,_injection_and_surjection
One-to-one correspondence
function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Given
Bijection
Representation of a mathematical function
the domain, and which set is the codomain. For example, to say that a function is onto (surjective) or not the codomain should be taken into account. The
Graph_of_a_function
Mathematical function
is, the functions of a real variable whose codomain is the set of real numbers. Nevertheless, the codomain of a function of a real variable may be any
Function_of_a_real_variable
Addition, multiplication, division, ...
produced is called the codomain, but the set of actual values attained by the operation is its codomain of definition, active codomain, image or range. For
Operation_(mathematics)
Function, homomorphism, or morphism
such as Serge Lang, use "function" only to refer to maps in which the codomain is a set of numbers (i.e. a subset of R or C), and reserve the term mapping
Map_(mathematics)
Set of the values of a function
inverse image (or preimage) of a given subset B {\displaystyle B} of the codomain Y {\displaystyle Y} is the set of all elements of X {\displaystyle X} that
Image_(mathematics)
Self-self morphism
operator theory. An endofunction is a function whose domain is equal to its codomain. A homomorphic endofunction is an endomorphism. Let S be an arbitrary set
Endomorphism
Theorems connecting continuity to closure of graphs
In mathematics, particularly in functional analysis, the closed graph theorem is a result connecting the continuity of a linear operator to a topological
Closed graph theorem (functional analysis)
Closed_graph_theorem_(functional_analysis)
"Pushed forward" from one measurable space to another
inverse image of the whole codomain is the whole domain, and the measure of the whole domain is 1, so the measure of the whole codomain is 1. This means that
Pushforward_measure
Function that returns its argument unchanged
is defined to be a function with X {\displaystyle X} as its domain and codomain, satisfying f ( x ) = x {\displaystyle f(x)=x} for all elements x {\displaystyle
Identity_function
Symbol representing a mathematical concept
concepts. In typed logic, F is a functional symbol with domain type T and codomain type U if, given any symbol X representing an object of type T, F(X) is
Function_symbol
Quotient space of a codomain of a linear map by the map's image
mapping of vector spaces f : X → Y is the quotient space Y / im(f) of the codomain of f by the image of f. The dimension of the cokernel is called the corank
Cokernel
Relationship between elements of two sets
domain with some elements of another set (possibly the same) called the codomain. Precisely, a binary relation over sets X {\displaystyle X} and Y {\displaystyle
Binary_relation
Algebraic element satisfying some of the criteria of an inverse
{\displaystyle A} is not surjective, then not all y {\displaystyle y} 's in its codomain have corresponding x {\displaystyle x} 's via A {\displaystyle A} . To
Generalized_inverse
Operation on mathematical functions
omitted. In a strict sense, the composition g ∘ f is only meaningful if the codomain of f equals the domain of g; in a wider sense, it is sufficient that the
Function_composition
Integer
an output codomain of every y ∈ Y from every input domain x ∈ X, there will be f−1(f(x)) = x, and f−1(f(y)) = y. When a subset of the codomain is specified
−1
Inputs at which function values are highest
arguments are defined over the domain of a function, the output is part of its codomain. Given an arbitrary set X {\displaystyle X} , a totally ordered set Y {\displaystyle
Arg_max
Ratio of polynomial functions
values of the variables for which the denominator is not zero, and the codomain is L. The set of rational functions over a field K is a field, the field
Rational_function
Mathematical concept
f^{-1}(y)={\frac {y+7}{5}}.} Let f be a function whose domain is the set X, and whose codomain is the set Y. Then f is invertible if there exists a function g from Y
Inverse_function
Principal square root of minus 1
domain to complex multiplication in the codomain. Real values in the domain represent scaling in the codomain (multiplication by a real scalar) with 1
Imaginary_unit
Topics referred to by the same term
Finnish name Irish National Teachers' Organisation Into, referring to the codomain of a mathematical functions, as in "F : A -> B" means F maps A into B Into
Into
Element mapped to itself by a mathematical function
domain and the codomain of f, and f(c) = c. In particular, f cannot have any fixed point if its domain is disjoint from its codomain. If f is defined
Fixed_point_(mathematics)
Most general completion of a commutative square given two morphisms with same codomain
diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. The pullback is written P = X ×f, Z, g Y. Usually the morphisms f and
Pullback_(category_theory)
Function with a smaller domain
{\displaystyle F} may be defined as a relation having domain A , {\displaystyle A,} codomain F {\displaystyle F} and graph G ( A ◃ R ) = { ( x , y ) ∈ F ( R ) : x ∈
Restriction_(mathematics)
Function's sensitivity to argument change
into some codomain (e.g. an n {\displaystyle n} -tuple of real numbers f ( x ) {\displaystyle f(x)} ), where both the domain and codomain are Banach
Condition_number
Finite ordered list of elements
:1\leq i\leq n\right\}} and with codomain codomain F = { a 1 , … , a n } , {\displaystyle \operatorname {codomain} F=\left\{a_{1},\ldots ,a_{n}\right\}
Tuple
Functions that send open (resp. closed) subsets to open (resp. closed) subsets
its codomain Y . {\displaystyle Y.} In fact, a relatively open map is a strongly open map if and only if its image is an open subset of its codomain. In
Open_and_closed_maps
Broad concept generalizing scalars in mathematics and physics
generally, has a domain of the same dimension (as a manifold) as its codomain, Conservative vector field, a vector field that is the gradient of a scalar
Vector (mathematics and physics)
Vector_(mathematics_and_physics)
Set of functions between two fixed sets
is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space
Function_space
behaviour under certain conditions. These properties concern the domain, the codomain and the image of functions. Injective function: has a distinct value for
List_of_types_of_functions
Fundamental trigonometric functions
(as well as those functions with the same function rule and domain whose codomain is a subset of R {\displaystyle \mathbb {R} } containing the interval [
Sine_and_cosine
Property of math operations which yield an inverse result when arguments' order reversed
arguments. The notion inverse refers to a group structure on the operation's codomain, possibly with another operation. Subtraction is an anticommutative operation
Anticommutative_property
Collection of mathematical objects
of f {\displaystyle f} , and B {\displaystyle B} is called the codomain of f {\displaystyle f} . The graph of a function f : A → B {\displaystyle
Set_(mathematics)
Last letter of the Greek alphabet
corresponding to the domain of a double integral. In topos theory, the (codomain of the) subobject classifier of an elementary topos. In combinatory logic
Omega
Generalization of a sequence of points
Moore–Smith sequence is a function whose domain is a directed set. The codomain of this function is usually some topological space. Nets directly generalize
Net_(mathematics)
Measure of local oscillation behavior
different concepts, related to the (local or global) structure of the codomain of a function or a measure. For a real-valued continuous function f, defined
Total_variation
Majorant and minorant in mathematics
functions. Given a function f with domain D and a preordered set (K, ≤) as codomain, an element y of K is an upper bound of f if y ≥ f(x) for each x in D.
Upper_and_lower_bounds
Kind of linear transformation
that its image f ( X ) {\displaystyle f(X)} is a bounded subset of its codomain. A linear map has this property if and only if it is identically 0. {\displaystyle
Bounded_operator
Mathematical concept for comparing objects
determines a partition on its domain, the set of preimages of singletons in the codomain. Thus an equivalence relation over X , {\displaystyle X,} a partition of
Equivalence_relation
Graphical representation of a morphism
{\text{cod}}:\Sigma _{1}\to \Sigma _{0}^{\star }} which assign a domain and codomain to each box, i.e. the input and output types. A morphism of monoidal signature
String_diagram
Point where function's value is zero
{\displaystyle \{0\}} in X {\displaystyle X} . Under the same hypothesis on the codomain of the function, a level set of a function f {\displaystyle f} is the zero
Zero_of_a_function
Notion of convergence in mathematics
\left(f_{n}\right)} all having the same domain X {\displaystyle X} and codomain Y {\displaystyle Y} is said to converge pointwise to a given function f
Pointwise_convergence
Maps whose domain and codomain are acted on by the same group, and the map commutes
spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, and when the function commutes
Equivariant_map
Applying operations to functions in terms of values for each input "point"
functions f {\displaystyle f} and g {\displaystyle g} with the same domain and codomain is defined by: ( f + g ) ( x ) = f ( x ) + g ( x ) . {\displaystyle (f+g)(x)=f(x)+g(x)
Pointwise
Function that takes one argument
argument. A unary operation is a special kind of unary function, whose codomain coincides with its domain. The successor function is a unary function.
Unary_function
Method of deriving conclusions
constructible Grothendieck Von Neumann Maps, cardinality Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering
Rule_of_inference
Topology on the real numbers
right-sided limit of f {\displaystyle f} at x {\displaystyle x} (when the codomain carries the standard topology) is the same as the usual limit of f {\displaystyle
Lower_limit_topology
Mathematical concept
element in the righthand class. It is not a function because its domain and codomain are not sets. In fact, the domain of the relation does not even need to
Transfinite_induction
Mathematical function, in linear algebra
a linear endomorphism, that is, a linear map with the same domain and codomain). Indeed, d d x ( a f ( x ) + b g ( x ) ) = a d f ( x ) d x + b d g ( x
Linear_map
Type of topology
and functional analysis. It was introduced by Ralph Fox in 1945. If the codomain of the functions under consideration has a uniform structure or a metric
Compact-open_topology
Map (arrow) between two objects of a category
Therefore, the source and the target of a morphism are often called domain and codomain respectively. Morphisms are equipped with a partial binary operation, called
Morphism
Symbolic boolean function representation, extension of BDDs
structure that is used to symbolically represent a Boolean function whose codomain is an arbitrary finite set S. An ADD is an extension of a reduced ordered
Algebraic_decision_diagram
Special mathematical function defined as sin(x)/x
&x=0\end{cases}}} Fields of application Signal processing, spectroscopy Domain, codomain and image Domain R {\displaystyle \mathbb {R} } Image [ − 0.217234 … ,
Sinc_function
Decision-making approach
human operator has ignored or answered to an insufficient manner. The codomain for the models that utilize adaptive collaborative control are queries
Adaptive collaborative control
Adaptive_collaborative_control
In linear algebra, relation between 3 dimensions
the linear map be finite-dimensional, there is no such assumption on the codomain. This means that there are linear maps not given by matrices for which
Rank–nullity_theorem
Model of the extended complex plane plus a point at infinity
meromorphic function can be thought of as a holomorphic function whose codomain is the Riemann sphere. In geometry, the Riemann sphere is the prototypical
Riemann_sphere
Approximating an arbitrary function with a well-behaved one
extrapolation, regression analysis, and curve fitting can be used. If the codomain (range or target set) of g is a finite set, one is dealing with a classification
Function_approximation
Distance-preserving mathematical transformation
{\displaystyle AA^{\dagger }=\operatorname {Id} _{V}} (i.e. the domain and codomain coincide and A {\displaystyle A} defines a coisometry). By the Mazur–Ulam
Isometry
Computation model defining an abstract machine
constructible Grothendieck Von Neumann Maps, cardinality Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering
Turing_machine
Type of non-sinusoidal waveform
Domain, codomain and image Domain R ∖ { n 2 } , n ∈ Z {\displaystyle \mathbb {R} \setminus \left\{{\tfrac {n}{2}}\right\},n\in \mathbb {Z} } Codomain { −
Square_wave_(waveform)
Branch of mathematics that studies sets
constructible Grothendieck Von Neumann Maps, cardinality Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering
Set_theory
Logarithm to the base of the mathematical constant e
Pure and applied mathematics Domain, codomain and image Domain R > 0 {\displaystyle \mathbb {R} _{>0}} Codomain R {\displaystyle \mathbb {R} } Image R
Natural_logarithm
Symbol representing a property or relation in logic
constructible Grothendieck Von Neumann Maps, cardinality Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering
Predicate_(logic)
Set-theoretic function
many instances, one can also construct a canonical inclusion into the codomain R → Y {\displaystyle R\to Y} known as the range of f . {\displaystyle f
Inclusion_map
Mathematical function, denoted exp(x) or e^x
complex function, which is a function with the complex numbers as domain and codomain, such that its restriction to the reals is the above-defined exponential
Exponential_function
Mathematical use of "for all"
{P}}Y\to {\mathcal {P}}X} between powersets, that takes subsets of the codomain of f back to subsets of its domain. The left adjoint of this functor is
Universal_quantification
Process of repeating items in a self-similar way
constructible Grothendieck Von Neumann Maps, cardinality Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering
Recursion
Technique used in mathematical logic
new element must belong to the domain of the extension, or to its image (codomain). As an example, the back-and-forth method can be used to prove Cantor's
Back-and-forth_method
Symbol connecting formulas in logic
constructible Grothendieck Von Neumann Maps, cardinality Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering
Logical_connective
Mathematical use of "there exists"
constructible Grothendieck Von Neumann Maps, cardinality Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering
Existential_quantification
Theorem about products in model theory
constructible Grothendieck Von Neumann Maps, cardinality Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering
Feferman–Vaught_theorem
Polynomial function of degree 3
function is a complex function that has the set of the complex numbers as its codomain, even when the domain is restricted to the real numbers. Setting f(x) =
Cubic_function
Machine learning framework
the lifting (lifting the codomain of the input function to a higher dimensional space) and projection (projecting the codomain of the intermediate function
Neural_operators
Mathematical set of all subsets of a set
constructible Grothendieck Von Neumann Maps, cardinality Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering
Power_set
Set whose elements all belong to another set
constructible Grothendieck Von Neumann Maps, cardinality Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering
Subset
Topics referred to by the same term
its domain is assigned to more than one elements of its codomain, and no element of its codomain is assigned to more than one element of its domain Cardinality
One-to-many
Logical connective OR
constructible Grothendieck Von Neumann Maps, cardinality Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering
Logical_disjunction
Proof in set theory
constructible Grothendieck Von Neumann Maps, cardinality Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering
Cantor's_diagonal_argument
Structure of a formal language
constructible Grothendieck Von Neumann Maps, cardinality Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering
Formal_grammar
Reasoning for mathematical statements
constructible Grothendieck Von Neumann Maps, cardinality Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering
Mathematical_proof
Mathematical set formed from two given sets
0th power of X is the singleton set, that has the empty function with codomain X as its unique element. Let Cartesian products be given A = A 1 × ⋯ ×
Cartesian_product
Non-sinusoidal waveform
codomain and image Domain R ∖ { n − 1 2 } , n ∈ Z {\displaystyle \mathbb {R} \setminus \left\{n-{\tfrac {1}{2}}\right\},n\in \mathbb {Z} } Codomain (
Sawtooth_wave
Diagram that shows all possible logical relations between a collection of sets
constructible Grothendieck Von Neumann Maps, cardinality Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering
Venn_diagram
Type of subroutine in computer science
unchanged. That is, a null function is an identity function whose domain and codomain are both the state space S {\displaystyle S} of the program, and for which:
Null_function
Function whose actual domain of definition may be smaller than its apparent domain
sets of functions defined on subsets of X {\displaystyle X} with same codomain Y {\displaystyle Y} : [ X ⇀ Y ] = ⋃ D ⊆ X [ D → Y ] , {\displaystyle [X\rightharpoonup
Partial_function
Non-sinusoidal waveform
application Electronics, synthesizers Domain, codomain and image Domain R {\displaystyle \mathbb {R} } Codomain [ − 1 , 1 ] {\displaystyle \left[-1,1\right]}
Triangle_wave
Mathematical function between groups that preserves multiplication structure
homomorphism that is surjective (or, onto); i.e., reaches every point in the codomain. Isomorphism A group homomorphism that is bijective; i.e., injective and
Group_homomorphism
Largest and smallest value taken by a function at a given point
arguments are defined over the domain of a function, the output is part of its codomain. Derivative test Infimum and supremum Limit superior and limit inferior
Maximum_and_minimum
In mathematics, a statement that has been proven
constructible Grothendieck Von Neumann Maps, cardinality Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering
Theorem
Limitative results in mathematical logic
constructible Grothendieck Von Neumann Maps, cardinality Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Complexity class used to classify decision problems
constructible Grothendieck Von Neumann Maps, cardinality Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering
NP_(complexity)
Non-contradiction of a theory
constructible Grothendieck Von Neumann Maps, cardinality Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering
Consistency
Criterion about convergence of series
A more general version of the Weierstrass M-test holds if the common codomain of the functions (fn) is a Banach space, in which case the premise | f
Weierstrass_M-test
CODOMAIN
CODOMAIN
CODOMAIN
CODOMAIN
Boy/Male
Hindu, Indian
Lucky
Boy/Male
African, Arabic, Gothic
King of Mask
Boy/Male
Muslim
Holiness, Sanctity
Boy/Male
Arabic, Muslim, Swahili
Beneficent; Charitable
Girl/Female
Australian, French, Greek, Latin
Defender of Mankind; Feminine of Alexander
Boy/Male
Gujarati, Hindu, Indian, Telugu
Good Natured
Boy/Male
American, Basque, Chinese, French, German
Of Mars; The God of War
Boy/Male
Indian, Punjabi, Sikh
Love for Babies
Surname or Lastname
English (of Norman origin) and French
English (of Norman origin) and French : topographic name from Old French molin ‘mill’.English (of Norman origin) : habitational name from a place in France called Moline(s).Swedish : ornamental name from mo ‘sandy heath’ + the common ornamental suffix -lin.In some cases, possibly Italian, a variant of Molino.
Boy/Male
Indian
Happiness
CODOMAIN
CODOMAIN
CODOMAIN
CODOMAIN
CODOMAIN