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Association of one output to each input
a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y
Function_(mathematics)
Function from sets to numbers
mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values
Set_function
Set-to-real map with diminishing returns
submodular set function (also known as a submodular function) is a set function that, informally, describes the relationship between a set of inputs and
Submodular_set_function
mathematics, a superadditive set function is a set function whose value when applied to the union of two disjoint sets is greater than or equal to the
Superadditive_set_function
subadditive set function is a set function whose value, informally, has the property that the value of function on the union of two sets is at most the
Subadditive_set_function
Mapping function
an additive set function is a function μ \mu mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum
Sigma-additive_set_function
Function whose values are sets (mathematics)
A set-valued function, also called a correspondence or set-valued relation, is a mathematical function that maps elements from one set, the domain of the
Set-valued_function
Order-preserving mathematical function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept
Monotonic_function
Set of all things that may be the input of a mathematical function
In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by dom ( f ) {\displaystyle \operatorname
Domain_of_a_function
Shape containing unit line segments in all directions
bounds on a circular maximal function analogous to the Kakeya maximal function. It was conjectured that there existed sets of measure zero containing a
Kakeya_set
primitive recursive set functions or primitive recursive ordinal functions are analogs of primitive recursive functions, defined for sets or ordinals rather
Primitive recursive set function
Primitive_recursive_set_function
Set of the values of a function
In mathematics, the image of a function f : X → Y {\displaystyle f:X\to Y} is the set of all f ( x ) {\displaystyle f(x)} such that x {\displaystyle
Image_(mathematics)
Point where function's value is zero
hypothesis on the codomain of the function, a level set of a function f {\displaystyle f} is the zero set of the function f − c {\displaystyle f-c} for some
Zero_of_a_function
Mathematical function characterizing set membership
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all
Indicator_function
Collection of mathematical objects
geometric shapes, variables, functions, or even other sets. Mathematics typically does not define precisely what constitutes a "set" or "collection", because
Set_(mathematics)
Real function with secant line between points above the graph itself
a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. In simple terms, a convex function graph
Convex_function
Fractal sets in complex dynamics of mathematics
set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function
Julia_set
Type of function in mathematics
an analytic function is a function that is locally represented by a convergent power series. More precisely, a real or complex function is analytic at
Analytic_function
Target set of a mathematical function
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 set Y in the notation
Codomain
Class of mathematical functions
function is a function on a lattice that, informally, has the property of being characterized by "increasing differences." Seen from the point of set
Supermodular_function
Continuous function that is not absolutely continuous
In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in
Cantor_function
Mathematical function such that every output has at least one input
surjective function (also known as surjection, or onto function /ˈɒn.tuː/) is a function f such that, for every element y of the function's codomain, there
Surjective_function
codomain of a function, the function does not change as a set since by definition it is just a set of ordered pairs. That is, a function does not determine
Implementation of mathematics in set theory
Implementation_of_mathematics_in_set_theory
Function whose actual domain of definition may be smaller than its apparent domain
In mathematics, a partial function f from a set X to a set Y is a function from a subset S of X (possibly the whole X itself) to Y. The subset S, that
Partial_function
Operation on mathematical functions
relations are true of composition of functions, such as associativity. Composition of functions on a finite set: If f = {(1, 1), (2, 3), (3, 1), (4, 2)}
Function_composition
Mathematical function with convex lower level sets
quasiconvex function is a real-valued function defined on a convex subset of a real vector space, such that for any real number y, the set of points on
Quasiconvex_function
Generalization of mass, length, area and volume
(cf. Dirac delta function) is given by δa(S) = χS(a), where χS is the indicator function of S . {\displaystyle S.} The measure of a set is 1 if it contains
Measure_(mathematics)
Kind of mathematical function
mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure
Measurable_function
Function that preserves distinctness
In mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements of its domain to distinct
Injective_function
Set of elements common to all of some sets
Hall. ISBN 0-13-181629-2. Rosen, Kenneth (2007). "Basic Structures: Sets, Functions, Sequences, and Sums". Discrete Mathematics and Its Applications (Sixth ed
Intersection_(set_theory)
Axiomatic set theories based on the principles of mathematical constructivism
total) function. This is often because the predicate in a case-wise would-be definition may not be decidable. Adopting the standard definition of set equality
Constructive_set_theory
Set of functions between two fixed sets
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which
Function_space
Mathematical concept
In mathematics, the inverse function of a function f (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists
Inverse_function
In geometry, set whose intersection with every line is a single line segment
smallest convex set containing A. A convex function is a real-valued function defined on an interval with the property that its epigraph (the set of points
Convex_set
Conceptual framework used in numerical analysis of surfaces and shapes
well-behaved boundary. Below it, the red surface is the graph of a level set function φ {\displaystyle \varphi } determining this shape, and the flat blue
Level-set_method
Function that is continuous everywhere but differentiable nowhere
concocted to challenge the notion that every continuous function is differentiable except on a set of isolated points. Weierstrass's demonstration that continuity
Weierstrass_function
Description of continuous random distribution
probability density function (PDF), density function, or simply density of an absolutely continuous random variable, is a function whose value at any given
Probability_density_function
Generalized mathematical function
It is a set-valued function with additional properties depending on context; though some authors do not distinguish between set-valued functions and multifunctions
Multivalued_function
Topics referred to by the same term
are sets and total functions, respectively Set (abstract data type), a data type in computer science that is a collection of distinct values Set (C++)
Set
One-to-one correspondence
bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the
Bijection
Mathematical function with no sudden changes
In mathematics, a continuous function is a function such that a small variation of its argument induces at most a small variation of its value. This implies
Continuous_function
Mathematical function that can be computed by a program
Computable functions are the basic objects of study in computability theory. Informally, a function is computable if there is an algorithm that computes
Computable_function
Ratio of polynomial functions
polynomial functions of x {\displaystyle x} and Q {\displaystyle Q} is not the zero function. The domain of f {\displaystyle f} is the set of all values
Rational_function
Maximum size of an independent set of the matroid
independent subset of S, and the rank function of the matroid maps sets of elements to their ranks. The rank function is one of the fundamental concepts
Matroid_rank
Function computable with bounded loops
§ Limitations below. The set of primitive recursive functions is known as PR in computational complexity theory. A primitive recursive function takes a fixed number
Primitive_recursive_function
Set of functions used to represent the electronic wave function
computational chemistry, a basis set is a set of functions (called basis functions) that is used to represent the electronic wave function in the Hartree–Fock method
Basis_set_(chemistry)
Mathematical set that can be enumerated
countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural
Countable_set
Zeta-like functions approximate arbitrary holomorphic functions
universality of zeta functions is the remarkable ability of the Riemann zeta function and other similar functions (such as the Dirichlet L-functions) to approximate
Zeta_function_universality
Subset of a function's domain on which its value is equal
In mathematics, a level set of a real-valued function f of n real variables is a set where the function takes on a given constant value c, that is: L
Level_set
Representation of a mathematical function
In mathematics, the graph of a function f {\displaystyle f} is the set of ordered pairs ( x , y ) {\displaystyle (x,y)} , where f ( x ) = y . {\displaystyle
Graph_of_a_function
Largest and smallest value taken by a function at a given point
maxima and minima of functions. As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded
Maximum_and_minimum
Javascript design pattern
are functions. let counter = (function () { let i = 0; return { get: function () { return i; }, set: function (val) { i = val; }, increment: function ()
Immediately invoked function expression
Immediately_invoked_function_expression
Constant function: has a fixed value regardless of its input. Empty function: whose domain equals the empty set. Set function: whose input is a set. Set-valued
List_of_types_of_functions
Mathematical set of all subsets of a set
indicator function or a characteristic function of a subset A of a set S with the cardinality |S| = n is a function from S to the two-element set {0, 1}
Power_set
Expressing a measure as an integral of another
between two measures defined on the same measurable space. A measure is a set function that assigns a consistent magnitude to the measurable subsets of a measurable
Radon–Nikodym_theorem
Subset of a function's codomain
image of a function are the same set; such a function is called surjective or onto. For any non-surjective function f : X → Y , {\displaystyle f:X\to
Range_of_a_function
Mapping arbitrary data to fixed-size values
A hash function is any function that can be used to map data of arbitrary size to fixed-size values, though there are some hash functions that support
Hash_function
Class of mathematical function
function on an open subset D {\displaystyle D} of the complex plane is a function that is holomorphic on all of D {\displaystyle D} except for a set of
Meromorphic_function
functions. They were introduced by René-Louis Baire in 1899. A Baire set is a set whose characteristic function is a Baire function. Baire functions of
Baire_function
exclusive possible worlds. The application of functions to value sets creates combinations of value sets from different worlds. Narrowing reduces those
Narrowing of algebraic value sets
Narrowing_of_algebraic_value_sets
Function uniquely mapping two numbers into a single number
a pairing function is a process to uniquely encode two natural numbers into a single natural number. Any pairing function can be used in set theory to
Pairing_function
Region with boundary of finite measure
measure. A synonym is set of (locally) finite perimeter. Basically, a set is a Caccioppoli set if its characteristic function is a function of bounded variation
Caccioppoli_set
About mathematical functions
invention of set theory by Georg Cantor, eventually led to the much more general modern concept of a function as a single-valued mapping from one set to another
History of the function concept
History_of_the_function_concept
Mathematical description of quantum state
In quantum mechanics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common
Wave_function
CPU instruction to set a memory location to a flag value and return its prior value
to 'initial' creates a new value (not just copying a reference). function TestAndSet(boolean_ref lock) { boolean initial = lock; lock = true; return initial;
Test-and-set
Smooth and compactly supported function
commonly used as cutoff functions, for example functions that are equal to 1 on a prescribed set and vanish outside a larger set, and as standard examples
Bump_function
Distance from origin of tangent hyperplanes
In mathematics, the support function hA of a non-empty closed convex set A in R n {\displaystyle \mathbb {R} ^{n}} describes the (signed) distances of
Support_function
Finite collection of distinct objects
function from a larger finite set to a smaller finite set. The natural numbers are defined abstractly by the Peano axioms, and can be constructed set-theoretically
Finite_set
Branch of mathematics that studies sets
function as a relation from one set (the domain) to another set (the range). Paul Halmos, Naive Set Theory, 1960, Springer Verlag. Thomas Jech, Set Theory
Set_theory
Statistical function that defines the quantiles of a probability distribution
the quantile function of a probability distribution is the inverse of its cumulative distribution function. That is, the quantile function of a distribution
Quantile_function
A set function is called fractionally subadditive, or XOS (not to be confused with OXS), if it is the maximum of several non-negative additive set functions
Fractionally subadditive valuation
Fractionally_subadditive_valuation
Evaluation of a function on its argument
In mathematics, function application (or evaluation) is the act of taking a function and an input from its domain to obtain the corresponding value from
Function_application
Smooth approximation of one-hot arg max
The softmax function, also known as softargmax or normalized exponential function, converts a tuple of K real numbers into a probability distribution
Softmax_function
Multiplicative function in number theory
The Möbius function μ ( n ) {\displaystyle \mu (n)} is a multiplicative function in number theory introduced by the German mathematician August Ferdinand
Möbius_function
Extended measure of size in mathematics
well-established for this set function, despite the fact that it is not a true measure in its modern definition, since Jordan-measurable sets do not form a σ-algebra
Peano–Jordan_measure
Function with a repeating pattern
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which are used to describe waves
Periodic_function
Egyptian god of the desert, storms, violence, and foreigners
possession of Horus's eye, when it appears on Set's head. Because Thoth is a moon deity in addition to his other functions, it would make sense, according to te Velde
Set_(deity)
Set function that is a precursor to a measure
In mathematics, a pre-measure is a set function that is, in some sense, a precursor to a bona fide measure on a given space. Indeed, one of the fundamental
Pre-measure
Mathematical function whose set of values is bounded
mathematics, a function f {\displaystyle f} defined on some set X {\displaystyle X} with real or complex values is called bounded if the set of its values
Bounded_function
Mathematical activation function in data analysis
set to 1) or trainable and "sigmoid" refers to the logistic function. The swish family was designed to smoothly interpolate between a linear function
Swish_function
Extension of the factorial function
gamma function (represented by Γ {\displaystyle \Gamma } , capital Greek letter gamma) is the most common extension of the factorial function to complex
Gamma_function
Function, homomorphism, or morphism
can be used interchangeably, but transformation often refers to a function from a set to itself. There are also a few less common uses in logic and graph
Map_(mathematics)
Mathematical function whose derivative exists
Banach states that the set of functions that have a derivative at some point is a meagre set in the space of all continuous functions. Informally, this means
Differentiable_function
Axiom of set theory
set X {\displaystyle X} of nonempty sets, there exists a choice function f {\displaystyle f} that is defined on X {\displaystyle X} and maps each set
Axiom_of_choice
Collection of sets in mathematics that can be defined based on a property of its members
also be generalised to classes. A class function is not a function in the usual sense, since it is not a set; it is rather a formula Φ ( x , y ) {\displaystyle
Class_(set_theory)
Comprehensive list of Magic: The Gathering card sets since its inception in 1993
similar function; however, they are always attached to a specific set or block, while compilations are free to pick and choose cards from any set. All expansion
List of Magic: The Gathering sets
List_of_Magic:_The_Gathering_sets
Type of mathematical function
value is c = 4. The domain of this function is the set of all real numbers. The image of this function is the singleton set {4}. The independent variable x
Constant_function
Study of mathematical algorithms for optimization problems
minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization
Mathematical_optimization
Approximating an arbitrary function with a well-behaved one
In general, a function approximation problem asks us to select a function that closely matches ("approximates") a function in a task-specific way.[better source needed]
Function_approximation
Complex-differentiable (mathematical) function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood
Holomorphic_function
Function that returns its argument unchanged
X {\displaystyle X} is a set, the identity function f {\displaystyle f} on X {\displaystyle X} is defined to be a function with X {\displaystyle X} as
Identity_function
Indicator function of rational numbers
mathematics, the Dirichlet function is the indicator function 1 Q {\displaystyle \mathbf {1} _{\mathbb {Q} }} of the set of rational numbers Q {\displaystyle
Dirichlet_function
Infinite set that is not countable
A set X is uncountable if and only if any of the following conditions hold: There is no injective function (hence no bijection) from X to the set of
Uncountable_set
Tasks in machine learning
training data set. The performance of the networks is then compared by evaluating the error function using an independent validation set, and the network
Training, validation, and test data sets
Training,_validation,_and_test_data_sets
Mathematical logic concept
set S is the range of a partial computable function. The set S is the range of a total computable function, or empty. If S is infinite, the function can
Computably_enumerable_set
Function returning one of only two values
In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually {true, false}, {0,1} or {−1,1})
Boolean_function
Field of medical research
Gain-of-function research (GoF research or GoFR) is medical research that genetically alters an organism in a way that may enhance the biological functions of
Gain-of-function_research
are two equivalent ways to define a cylinder set measure. One way is to define it directly as a set function on the cylindrical algebra such that certain
Cylinder_set_measure
Function that applies a set to itself
transformation, transform, or self-map is a function f, usually with some geometrical underpinning, that maps a set X to itself, i.e. f: X → X. Examples include
Transformation_(function)
Function family in complex analysis
a holomorphic function on an open set D {\displaystyle D} , then f ( z ¯ ) {\displaystyle f({\bar {z}})} is an antiholomorphic function on D ¯ {\displaystyle
Antiholomorphic_function
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