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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
Properties of mathematical functions
g\circ f} is injective, then it can only be concluded that f {\displaystyle f} is injective (see figure). Every embedding is injective. A function is surjective
Bijection, injection and surjection
Bijection,_injection_and_surjection
Inclusion of one mathematical structure in another, preserving properties of interest
continuously differentiable function to be (among other things) locally injective. Every fiber of a locally injective function f:X→Y is necessarily a discrete
Embedding
Function whose actual domain of definition may be smaller than its apparent domain
partial function which is injective. An injective partial function may be inverted to an injective partial function, and a partial function which is
Partial_function
Mathematical function such that every output has at least one input
be unique; the function f may map one or more elements of X to the same element of Y. The term surjective and the related terms injective and bijective
Surjective_function
Mathematical concept
all x∈X implies that f is injective, and the condition f(g(y))=y for all y∈Y implies that f is surjective. The inverse function f −1 to f can be explicitly
Inverse_function
Differentiable function whose derivative is everywhere injective
function f itself need not be injective, only its derivative must be. A related concept is that of an embedding. A smooth embedding is an injective immersion
Immersion_(mathematics)
Association of one output to each input
element. An empty function is always injective. If X is not the empty set, then f is injective if and only if there exists a function g : Y → X {\displaystyle
Function_(mathematics)
Operation on mathematical functions
composition of one-to-one (injective) functions is always one-to-one. Similarly, the composition of onto (surjective) functions is always onto. It follows
Function_composition
Systematic classification of 12 related enumerative problems concerning two finite sets
equivalent to counting injective functions N → X. Counting n-combinations of X is equivalent to counting injective functions N → X up to permutations
Twelvefold_way
One-to-one correspondence
to be "one-to-one functions" and are called injections (or injective functions). With this terminology, a bijection is a function which is both a surjection
Bijection
Theorem in combinatorics
A to B that is not injective, then no surjection from A to B is injective. In fact no function of any kind from A to B is injective. This is not true for
Pigeonhole_principle
Mathematical set that can be enumerated
natural numbers. Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in
Countable_set
Theorem in mathematics
space, is a local homeomorphism that is injective on A {\displaystyle A} , then f {\displaystyle f} is injective on some neighborhood of A {\displaystyle
Inverse_function_theorem
Function with a smaller domain
(respectively, a continuous map, etc.). The restriction of the non-injective function f : R → R , x ↦ x 2 {\displaystyle f:\mathbb {R} \to \mathbb {R}
Restriction_(mathematics)
Function that returns its argument unchanged
{\displaystyle X} . The identity function on X {\displaystyle X} is clearly an injective function as well as a surjective function (its codomain is also its
Identity_function
Theorem in set theory
that, if there exist injective functions f : A → B and g : B → A between the sets A and B, then there exists a bijective function h : A → B. In terms of
Schröder–Bernstein_theorem
Topics referred to by the same term
(medicine), insertion of liquid into the body with a syringe Injective function, a mathematical function mapping distinct arguments to distinct values Injection
Injection
Hash function without any collisions
an injective function. Perfect hash functions may be used to implement a lookup table with constant worst-case access time. A perfect hash function can
Perfect_hash_function
Method to measure temperature quantitatively
{c}}} (the cardinality of the continuum), then one can construct an injective function f : M → R {\displaystyle f\colon M\to \mathbb {R} } into the real
Scale_of_temperature
Injective homomorphism
Monomorphisms are a categorical generalization of injective functions (also called "one-to-one functions"); in some categories the notions coincide, but
Monomorphism
Size of a set in mathematics
called injective. If a function covers every member in the output set, it is called surjective. If a function is both injective and surjective, it is called
Cardinality
Finite collection of distinct objects
the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set. The natural numbers
Finite_set
Smooth map which is a diffeomorphism upon restriction
smooth immersion is a locally injective function, while invariance of domain guarantees that any continuous injective function between manifolds of equal
Local_diffeomorphism
Test for the injectivity of a function
whether a function is injective (i.e., one-to-one). A horizontal line is a straight, flat line that goes from left to right. Given a function f : R → R
Horizontal_line_test
Cryptocurrency
Injective (INJ) is a blockchain optimized for Web3 finance. Injective was launched by Injective Labs, which was founded by Eric Chen and Albert Chon in
Injective_(blockchain)
continuous equivariant injective function f : S G → S G {\displaystyle f:S^{G}\to S^{G}} is also surjective. The implication from injectivity to surjectivity
Surjunctive_group
Infinite set that is not countable
if and only if any of the following conditions hold: There is no injective function (hence no bijection) from X to the set of natural numbers. X is nonempty
Uncountable_set
Topics referred to by the same term
Simple path may refer to: Simple curve, a continuous injective function from an interval in the set of real numbers R {\displaystyle \mathbb {R} } to R
Simple_path
Mathematical concept
analysis, a holomorphic function on an open subset of the complex plane is called univalent if it is injective. The function f : z ↦ 2 z + z 2 {\displaystyle
Univalent_function
one-way function, the Jacobi symbol of the preimage can be easily computed from that of the image. It is clear that if an injective function has a hard-core
Hard-core_predicate
non-injective computations within a reversible framework. The core idea is to transform a non-injective function f: X → Y into a related injective function
Program Inversion, Interpretation, and Injectivization
Program_Inversion,_Interpretation,_and_Injectivization
Fuel injection system for diesel engines
A unit injector (UI) is a high-pressure integrated direct fuel injection system for diesel engines, combining the injector nozzle and the injection pump
Unit_injector
Technique in integral evaluation
functions. A bi-Lipschitz function is a Lipschitz function φ : U → Rn which is injective and whose inverse function φ−1 : φ(U) → U is also Lipschitz. By Rademacher's
Integration_by_substitution
Set-theoretic function
use this hooked arrow for any embedding.) This and other analogous injective functions from substructures are sometimes called natural injections. Given
Inclusion_map
Family of higher-order functions
reduce, accumulate, aggregate, compress, or inject. Typically, a fold is presented with a combining function, a top node of a data structure, and possibly
Fold_(higher-order_function)
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 is
Function_space
Functions of an angle
\end{aligned}}} The trigonometric functions are periodic, and hence not injective, so strictly speaking, they do not have an inverse function. However, on each interval
Trigonometric_functions
Order-preserving mathematical function
an inverse function on T {\displaystyle T} for f {\displaystyle f} . In contrast, each constant function is monotonic, but not injective, and hence cannot
Monotonic_function
Type of metric space
the equivalence between hyperconvexity and injectivity, these spaces are all also injective. In an injective space, the radius of the minimum ball that
Injective_metric_space
Every set is smaller than its power set
X} and Y {\displaystyle Y} if and only if there is an injective function but no bijective function from X {\displaystyle X} to Y {\displaystyle Y} . It
Cantor's_theorem
image of functions. Injective function: has a distinct value for each distinct input. Also called an injection or, sometimes, one-to-one function. In other
List_of_types_of_functions
(Mathematical) decomposition into a product
objects. For example, every function may be factored into the composition of a surjective function with an injective function. Matrices possess many kinds
Factorization
Number taken as representative of a list of numbers
{\displaystyle f} -mean, also known as the quasi-arithmetic mean. For an injective function f : I → R {\displaystyle f\colon I\rightarrow \mathbb {R} } on an
Average
Generalized mathematical function
are all examples of multivalued functions that come about from non-injective functions. Since the original functions do not preserve all the information
Multivalued_function
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
Function that takes two inputs
This example is injective in each input separately, because the functions f x and f y are always injective. However, it's not injective in both variables
Binary_function
Mathematical object in sheaf cohomology
example the Leray spectral sequence. An injective sheaf F {\displaystyle {\mathcal {F}}} is a sheaf that is an injective object of the category of abelian sheaves;
Injective_sheaf
Type of mathematical function
numbering is an injective function mapping the set of well-formed formulae (a finite sequence of symbols on which the ordinal notation function is defined)
Ordinal_notation
Complex analysis function
A Nevanlinna function maps the upper half-plane to itself or a real constant, but is not necessarily injective or surjective. Functions with this property
Nevanlinna_function
Hash function that is suitable for use in cryptography
A cryptographic hash function (CHF) is a hash algorithm (a map of an arbitrary binary string to a binary string with a fixed size of n {\displaystyle n}
Cryptographic_hash_function
Size of a possibly infinite set
numbers is defined as follows: |X| ≤ |Y| means that there exists an injective function from X to Y. The Cantor–Bernstein–Schroeder theorem states that if
Cardinal_number
Set with an equinumerous proper subset
some proper subset of A; there is a function A → A that is injective but not surjective; there is an injective function A ∪ {A} → A; it is dually Dedekind-infinite
Dedekind-infinite_set
Comparison in computer science
by proving the existence of a particular injective function. For profit maximization problems, the function can be any one-to-one mapping from elements
Charging_argument
Multivalued function in mathematics
entire negative real axis. The functions W k ( z ) , k ∈ Z {\displaystyle W_{k}(z),k\in \mathbb {Z} } are all injective and their ranges are disjoint.
Lambert_W_function
Well-quasi-ordering of finite trees
T_{2}} and write T 1 ≤ T 2 {\displaystyle T_{1}\leq T_{2}} if there is an injective map F {\displaystyle F} from the vertices of T 1 {\displaystyle T_{1}}
Kruskal's_tree_theorem
Ordered listing of items in collection
only if there exists an injective function from it into the natural numbers. The natural numbers are enumerable by the function f(x) = x. In this case
Enumeration
Mathematical ranking of a set
class on X . {\displaystyle X.} However, the function f {\displaystyle f} induces an injective function that maps the partition on X {\displaystyle X}
Weak_ordering
one containing all members of some type vector 1. Essentially an injective function from a class to itself (for example, a vector in a vector space acting
Glossary of Principia Mathematica
Glossary_of_Principia_Mathematica
Retroazimuthal map projection
the back and front hemispheres overlap, making the projection a non-injective function. The back hemisphere can be rotated 180° to avoid overlap, but in
Hammer retroazimuthal projection
Hammer_retroazimuthal_projection
Category theory generalization of fumction factorization
can be shown that every function can be written as the composite of a surjective function followed by an injective function. Factorization systems are
Factorization_system
Functions that send open (resp. closed) subsets to open (resp. closed) subsets
invariance of domain theorem states that a continuous and locally injective function between two n {\displaystyle n} -dimensional topological manifolds
Open_and_closed_maps
Type of topological space
notation of numbers. (See Cantor space.) Every fiber of a locally injective function is necessarily a discrete subspace of its domain. In the foundations
Discrete_space
Programming technique of receiving dependencies
Instead, the receiving "client" (object or function) is provided with its dependencies by external code (an "injector"), which it is not aware of. Dependency
Dependency_injection
Continuous deformation between two continuous functions
straight line. The problem is that, while continuous, this is not an injective function of the Euclidean space that the knot is embedded in. An ambient isotopy
Homotopy
Generalization of linear assignment problem from two to multiple dimensions
assignment of agents to each job characteristic is an injective function, or one-to-one function from agents to a given job characteristic. Alternatively
Multidimensional assignment problem
Multidimensional_assignment_problem
Normal form used in database normalization
in the structure of that table. Attribute-value system Bijection Injective function First normal form (1NF) Second normal form (2NF) Third normal form
Fourth_normal_form
Result in combinatorics and graph theory
can be formalized by defining a transversal to be the image of an injective function f : F → X {\displaystyle f:{\mathcal {F}}\to X} such that f ( S )
Hall's_marriage_theorem
Mathematical function that outputs real values
In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each
Real-valued_function
Study of computable functions and Turing degrees
one-one reducible (or 1-reducible) to B if there is a total computable injective function f such that each n is in A if and only if f(n) is in B. Many-one reducibility:
Computability_theory
Subset that is closed and has no isolated points
locally compact Hausdorff space with no isolated points, there is an injective function (not necessarily continuous) from the Cantor space to X, and so X
Perfect_set
Euclidean space without distance and angles
a natural injective function from an affine space into the set of prime ideals (that is the spectrum) of its ring of polynomial functions. When affine
Affine_space
Topics referred to by the same term
may refer to: One-to-one function, also called an injective function One-to-one correspondence, also called a bijective function One-to-one (communication)
One-to-one
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
function that maps distinct elements of its domain to distinct elements of its codomain, also known as an injective function or one-to-one function.
Glossary_of_logic
Mathematical functions
x {\displaystyle x} -element set, or equivalently the number of injective functions from a set of size n {\displaystyle n} to a set of size x {\displaystyle
Falling_and_rising_factorials
Set of marks along a ruler such that no two pairs of marks are the same distance apart
Such a form can be achieved through translation and reflection. An injective function f : { 1 , 2 , . . . , m } → { 0 , 1 , . . . , n } {\displaystyle f:\left\{1
Golomb_ruler
Mathematical function revertible near each point
if it is a locally injective map (meaning that every point in U has a neighborhood N such that the restriction of f to N is injective). Local homeomorphisms
Local_homeomorphism
Degree of differentiability of a function or map
transform gives an injective (but not surjective) map Ck(M)→C0(M). The above spaces occur naturally in applications where functions having derivatives
Smoothness
Injective polynomial functions are bijective
often given as this special case: If P {\displaystyle P} is an injective polynomial function from an n {\displaystyle n} -dimensional complex vector space
Ax–Grothendieck_theorem
Database normalization
candidate key as the table T is excluded. Attribute–value system Injective function First normal form (1NF) Second normal form (2NF) Third normal form
Fifth_normal_form
Order whose elements are all comparable
(more strongly, these are well-orders). If X is any set and f an injective function from X to a totally ordered set then f induces a total ordering on
Total_order
Study of mathematical knots
"nearly" injective and continuous function K : [ 0 , 1 ] → R 3 {\displaystyle K\colon [0,1]\to \mathbb {R} ^{3}} , with the only "non-injectivity" being
Knot_theory
Type of mathematical function
mathematics, a constant function is a function whose (output) value is the same for every input value. As a real-valued function of a real-valued argument
Constant_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
Mathematical function
mathematics, a function of a real variable is a function whose domain is a subset of R {\displaystyle \mathbb {R} } . Many real functions that are often
Function_of_a_real_variable
Overview of and topical guide to discrete mathematics
mathematical function Range of a function – Subset of a function's codomain Image (mathematics) – Set of the values of a function Injective function – Function that
Outline of discrete mathematics
Outline_of_discrete_mathematics
Structure-preserving map between two algebraic structures of the same type
then f {\displaystyle f} is bijective. In fact, f {\displaystyle f} is injective, as f ( x ) = f ( y ) {\displaystyle f(x)=f(y)} implies x = g ( f ( x
Homomorphism
Axiomatic logical system
facts about N (it is an infinite set bounded by 0) and S (it is an injective function whose domain is N) needed for non-triviality. The converse of (2)
Robinson_arithmetic
Set theory concept
cardinals, for each x ∈ β {\displaystyle x\in \beta } , choose an injective function f x : x → α {\displaystyle f_{x}:x\to \alpha } and define the sets
Measurable_cardinal
Binary relation over a set and itself
called a partial function. A (total) function is a partial function that is left-total. An injective function (or partial function) is one whose inverse
Homogeneous_relation
Alternative mathematical ordering
called an embedding if it is both monotone and injective.[nb] Equivalently, an embedding is a function that pushes forward the ordering on X: whenever
Cyclic_order
Mathematical function with multiple real-number arguments
In mathematics, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being
Function of several real variables
Function_of_several_real_variables
Certain kind of cardinal number in set theory
\hookrightarrow X\}} be the class of all ordinal numbers β for which an injective function exists from β into X. First, we verify that α is a set. X × X is a
Hartogs_number
Type of mathematical expression
to R[x] sending r to itself considered as a constant polynomial is an injective ring homomorphism, by which R is viewed as a subring of R[x]. In particular
Polynomial
Azimuth angle of the Sun's position
in The Nautical Almanac. Since sine, cosine and tangent are not injective functions, the inverse of each does not give a unique correct solution, and
Solar_azimuth_angle
On converting relations to functions of several real variables
{\textstyle {\frac {\partial f}{\partial y}}(x_{0},y_{0})\neq 0} ), thus it is injective. If φ , ϕ {\textstyle \varphi ,\phi } are solutions to the differential
Implicit_function_theorem
Generalization of additive and multiplicative inverses
is even or odd. More generally, a function has a left inverse for function composition if and only if it is injective, and it has a right inverse if and
Inverse_element
Mathematical transform that expresses a function of time as a function of frequency
takes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output
Fourier_transform
Functors which are surjective and injective on hom-sets
In category theory, a faithful functor is a functor that is injective on hom-sets, and a full functor is surjective on hom-sets. A functor that has both
Full_and_faithful_functors
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INJECTIVE FUNCTION
INJECTIVE FUNCTION
INJECTIVE FUNCTION
INJECTIVE FUNCTION
INJECTIVE FUNCTION
INJECTIVE FUNCTION
INJECTIVE FUNCTION
INJECTIVE FUNCTION
INJECTIVE FUNCTION
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