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Identity function

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Graph of the identity function on the real numbers

In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged. That is, when $f$ is the identity function, the equality $f (x) = x$ is true for all values of $x$ to which $f$ can be applied.

Definition

[edit]

Formally, if $X$ is a set, the identity function $f$ on $X$ is defined to be a function with $X$ as its domain and codomain, satisfying

f (x) = x

for all elements

x

X

.^[1]

In other words, the function value $f (x)$ in the codomain $X$ is always the same as the input element $x$ in the domain $X$ . The identity function on $X$ is clearly an injective function as well as a surjective function (its codomain is also its range), so it is bijective.^[2]

The identity function $f$ on $X$ is often denoted by $id X$ .

In set theory, where a function is defined as a particular kind of binary relation, the identity function is given by the identity relation, or diagonal of $X$ .^[3]

Algebraic properties

[edit]

If $f : X \to Y$ is any function, then $f \circ id X = f = id Y \circ f$ , where "∘" denotes function composition.^[4] In particular, $id X$ is the identity element of the monoid of all functions from $X$ to $X$ (under function composition).

Since the identity element of a monoid is unique,^[5] one can alternately define the identity function on $M$ to be this identity element. Such a definition generalizes to the concept of an identity morphism in category theory, where the endomorphisms of $M$ need not be functions.

Properties

[edit]

The identity function is a linear operator when applied to vector spaces.^[6]
In an $n$ -dimensional vector space the identity function is represented by the identity matrix $I n$ , regardless of the basis chosen for the space.^[7]
The identity function on the positive integers is a completely multiplicative function (essentially multiplication by 1), considered in number theory.^[8]
In a metric space the identity function is trivially an isometry. An object without any symmetry has as its symmetry group the trivial group containing only this isometry (symmetry type $C 1$ ).^[9]
In a topological space, the identity function is always continuous.^[10]
The identity function is idempotent.^[11]

References

[edit]

^ Knapp, Anthony W. (2006). Basic algebra. Springer. ISBN 978-0-8176-3248-9.
^ Mapa, Sadhan Kumar (7 April 2014). Higher Algebra Abstract and Linear (11th ed.). Sarat Book House. p. 36. ISBN 978-93-80663-24-1.
^ Proceedings of Symposia in Pure Mathematics. American Mathematical Society. 1974. p. 92. ISBN 978-0-8218-1425-3. ...then the diagonal set determined by M is the identity relation...
^ Nel, Louis (2016). Continuity Theory. Cham: Springer. p. 21. doi:10.1007/978-3-319-31159-3. ISBN 978-3-319-31159-3.
^ Rosales, J. C.; García-Sánchez, P. A. (1999). Finitely Generated Commutative Monoids. Nova Publishers. p. 1. ISBN 978-1-56072-670-8. The element 0 is usually referred to as the identity element and if it exists, it is unique
^ Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International
^ T. S. Shores (2007). Applied Linear Algebra and Matrix Analysis. Undergraduate Texts in Mathematics. Springer. ISBN 978-038-733-195-9.
^ D. Marshall; E. Odell; M. Starbird (2007). Number Theory through Inquiry. Mathematical Association of America Textbooks. Mathematical Assn of Amer. ISBN 978-0883857519.
^ Anderson, James W. (2007). Hyperbolic geometry. Springer undergraduate mathematics series (2. ed., corr. print ed.). London: Springer. ISBN 978-1-85233-934-0.
^ Conover, Robert A. (2014-05-21). A First Course in Topology: An Introduction to Mathematical Thinking. Courier Corporation. p. 65. ISBN 978-0-486-78001-6.
^ Conferences, University of Michigan Engineering Summer (1968). Foundations of Information Systems Engineering. we see that an identity element of a semigroup is idempotent.

v t e Function
History List of specific functions
Types by domain and codomain	$X \to 𝔹$ $𝔹 \to X$ $𝔹ⁿ \to X$ $X \to ℤ$ $ℤ \to X$ $X \to ℝ$ $ℝ \to X$ $ℝⁿ \to X$ $X \to ℂ$ $ℂ \to X$ $ℂⁿ \to X$
Classes/properties	Constant Identity Linear Polynomial Rational Algebraic Analytic Smooth Continuous Measurable Injective Surjective Bijective
Constructions	Restriction Composition λ Inverse
Generalizations	Relation (Binary relation) Set-valued Multivalued Partial Implicit Space Higher-order Morphism Functor
Category

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In mathematics, the identity function, also known as the identity map or identity transformation, is a function that leaves unchanged every element of its domain, mapping each input directly to itself.^[1] For a set

A

, the identity function is denoted

\mathrm{id}_A: A \to A

and defined by

\mathrm{id}_A(a) = a

for all

a \in A

.^[2] This makes it the simplest possible function from a set to itself, serving as a foundational concept in set theory, algebra, and analysis.^[3] The identity function exhibits several key properties that underscore its importance across mathematical disciplines. It is bijective, meaning it is both injective (one-to-one) and surjective (onto), as every element in the codomain has exactly one preimage, which is itself.^[4] Additionally, it acts as the identity element under function composition: for any function

f: A \to B

, composing

f

with

\mathrm{id}_A

on the left or

\mathrm{id}_B

on the right yields

f

unchanged.^[5] The identity function is also an involution, as its own inverse, satisfying

\mathrm{id}_A \circ \mathrm{id}_A = \mathrm{id}_A

.^[4] In the context of real numbers, the identity function is often expressed as

f(x) = x

, whose graph is the straight line

y = x

with slope 1 and y-intercept 0, defined over all real numbers with both domain and range

\mathbb{R}

.^[6] This form is linear and strictly increasing, preserving the order of real numbers.^[6] Beyond sets and reals, the identity function generalizes to other structures, such as vector spaces where it is the identity linear transformation, or in category theory as the identity morphism, facilitating the study of isomorphisms and equivalences.^[4]

Definition and Basic Concepts

Formal Definition

In mathematics, the identity function on a set

X

is defined as the function

f: X \to X

such that

f(x) = x

for every

x \in X

. This construction applies to any set

X

, where the domain and codomain are identical.^[1] From its definition, the identity function is inherently a bijection, as it is both injective and surjective.^[7] Injectivity follows because if

f(x) = f(y)

, then

x = y

; surjectivity holds because for every

y \in X

, the element

x = y

satisfies

f(x) = y

.^[8] A specific instance is the identity function on the real numbers, denoted

\mathrm{id}: \mathbb{R} \to \mathbb{R}

and defined by

\mathrm{id}(x) = x

for all

x \in \mathbb{R}

.^[9]

Notation and Representations

The identity function on a set

X

is commonly denoted by

\mathrm{id}_X

, where

\mathrm{id}_X: X \to X

satisfies

\mathrm{id}_X(x) = x

for all

x \in X

.^[10] This notation emphasizes the set-specific nature of the function and is standard in set theory and category theory contexts.^[11] In linear algebra, the identity function is often represented as the identity operator

I

, acting on a vector space

V

such that

I \mathbf{v} = \mathbf{v}

for any vector

\mathbf{v} \in V

. This operator is realized concretely as the identity matrix

I_n

n

-dimensional space, an

n \times n

square matrix with 1s on the main diagonal and 0s elsewhere. The entries of

I_n

can be expressed using the Kronecker delta

\delta_{ij}

, defined as

\delta_{ij} = 1

i = j

and

\delta_{ij} = 0

otherwise, so that

(I_n)_{ij} = \delta_{ij}

.^[12] For the identity function on the real numbers,

f: \mathbb{R} \to \mathbb{R}

with

f(x) = x

, its graphical representation is the straight line

y = x

in the Cartesian plane, passing through the origin with a slope of 1.^[13] This line visually captures the function's property of mapping each input to itself without alteration.^[14]

Algebraic and Functional Properties

Algebraic Properties

In the category of sets, the set of all functions from a set

X

to itself forms a monoid under function composition, where the identity function

\mathrm{id}_X

, defined by

\mathrm{id}_X(x) = x

for all

x \in X

, serves as the multiplicative identity element.^[15] For any function

f: X \to X

, the composition satisfies

f \circ \mathrm{id}_X = \mathrm{id}_X \circ f = f

, ensuring that the identity function acts as a neutral element in this algebraic structure.^[16] This property highlights the identity function's central role in preserving the structure of compositions without altering the input-output behavior of any other function. The identity function exhibits the algebraic trait of being an involution under composition, meaning

\mathrm{id}_X \circ \mathrm{id}_X = \mathrm{id}_X

.^[17] Consequently, it is its own inverse, as the left and right compositions with itself yield the identity, distinguishing it as a self-inverse element in the monoid.^[18] In this monoid of functions under composition, the identity function is the unique element that serves simultaneously as both a left inverse and a right inverse to itself, underscoring its idempotent nature and foundational position in algebraic manipulations involving function chains.^[19] This commutativity extends universally: the identity function commutes with every function in the monoid, as

\mathrm{id}_X \circ f = f \circ \mathrm{id}_X = f

holds for all

f

, reflecting its neutral compatibility in algebraic operations.^[20] In the specific context of linear algebra, where linear maps between vector spaces are represented by matrices, the identity function corresponds to the identity matrix

I

, satisfying

I A = A I = A

for any matrix

A

compatible with the dimensions, thereby acting as the identity for matrix multiplication, which mirrors composition of linear transformations.^[21]

Functional and Analytic Properties

The identity function

id: \mathbb{R} \to \mathbb{R}

, defined by

id(x) = x

, is continuous at every point in its domain. This follows from the definition of continuity, as the limit of

id(x)

x

approaches any point

a \in \mathbb{R}

equals

a

, matching the function value at that point.^[22] Moreover, it is differentiable everywhere on

\mathbb{R}

, with first derivative

id'(x) = 1

. The identity function is infinitely differentiable, as higher-order derivatives satisfy

id''(x) = 0

and all subsequent derivatives are zero, reflecting its linear nature as a polynomial of degree one.^[22]^[23] The identity function is strictly increasing on

\mathbb{R}

, since for any

x < y

, it holds that

id(x) = x < y = id(y)

. This strict monotonicity implies that the function is injective, as distinct inputs produce distinct outputs.^[22]^[24] Every point in the domain serves as a fixed point of the identity function, satisfying

id(x) = x

for all

x \in \mathbb{R}

. This property underscores its role as the neutral element in function composition. In the context of metric spaces, the identity function acts as an isometry, preserving distances such that

d(id(x), id(y)) = d(x, y)

for all

x, y

in the space. This preservation holds trivially by the definition of the metric and confirms its bijectivity onto itself.^[25]

Applications and Generalizations

In linear algebra, the identity matrix

I

plays a central role in solving systems of linear equations, where the equation

I \mathbf{x} = \mathbf{b}

directly implies

\mathbf{x} = \mathbf{b}

, serving as the trivial case that underscores the multiplicative property of the identity.^[26] This property extends to more complex methods like Gauss-Jordan elimination, where row operations transform the coefficient matrix to

I

, yielding the solution vector.^[27] Additionally, the identity matrix forms the basis for projection operators, representing the orthogonal projection onto the entire vector space, as

I \mathbf{v} = \mathbf{v}

for any vector

\mathbf{v}

, and enabling derivations of projections onto subspaces via

I - P

where

P

is a subspace projection.^[28] In category theory, the identity morphism

\mathrm{id}_A: A \to A

for each object

A

is a foundational element, ensuring that every category has morphisms that act as neutral elements under composition, satisfying

\mathrm{id}_A \circ f = f = f \circ \mathrm{id}_A

for any morphism

f

with appropriate domain and codomain.^[29] This identity is essential to key definitions, such as those of functors, natural transformations, and limits, where it guarantees the existence of canonical arrows that preserve structure across categories.^[30] In computer science, particularly functional programming, the identity function—often implemented as a no-op like Python's lambda x: x or OCaml's let id x = x—serves as a polymorphic utility for operations such as mapping over data structures without alteration or as a placeholder in higher-order functions.^[31] It facilitates testing function compositions, enabling verification that

f \circ \mathrm{id} = f

, and is integral to concepts like monads and applicative functors where it acts as the neutral element.^[32] In physics, the identity transformation represents the unchanged state in symmetry operations, forming the neutral element in symmetry groups that describe conserved quantities under physical laws, such as rotations or translations in classical mechanics.^[33] In particle physics, it is the identity element in Lie groups like the Lorentz group, essential for classifying particles and interactions via representations where the identity operator

U(0) = I

corresponds to no transformation, underpinning gauge symmetries and conservation principles.^[34] In abstract algebra, the identity function, known as the identity homomorphism between a group or ring and itself, maps every element to itself while preserving the algebraic operations, providing a baseline for analyzing non-trivial structures.^[35] This role is crucial in proofs of isomorphism theorems and kernel properties, such as demonstrating that its kernel is trivial (containing only the identity element) or verifying uniqueness of certain homomorphisms.^[36]

Generalizations to Other Structures

In algebraic structures such as rings, the concept of the identity extends to the multiplicative identity element, denoted 1, which satisfies

1 \cdot r = r \cdot 1 = r

for every element

r

in the ring.^[37] This element ensures that the ring has a unit with respect to multiplication, distinguishing unital rings from rngs (rings without identity). In modules over a unital ring

R

, the scalar multiplication by the identity

1_R

acts as the identity on the module, meaning

1_R \cdot m = m

for all module elements

m

, preserving the module's structure under ring actions.^[38] In topology, the identity map

id_X: X \to X

, defined by

id_X(x) = x

for all

x \in X

, serves as a homeomorphism between a topological space

X

and itself, as it is a continuous bijection with a continuous inverse (itself).^[39] This property underscores the identity map's role in demonstrating that any topological space is homeomorphic to itself, facilitating comparisons of topological invariants without altering the space's structure. In type theory and logic, identity manifests as identity types, which formalize equality relations between terms of the same type. In Martin-Löf intensional type theory, the identity type

Id_A(a, b)

between elements

a, b: A

captures proofs of equality, with reflexivity providing the constructor

\mathsf{refl}_a: Id_A(a, a)

.^[40] In homotopy type theory (HoTT), these identity types acquire a richer path-like structure, interpreting equalities as paths in an

\infty

-groupoid, where higher identities represent homotopies between paths, enabling univalent foundations that treat isomorphic types as equal.^[40] Partial identities arise as restrictions of the full identity function to a subset of the domain, functioning as the identity on that subset while undefined elsewhere, akin to partial functions in set theory. For instance, given a set

X

and subset

S \subseteq X

, the partial identity on

S

maps each

s \in S

to itself, with domain

S

, and can be represented as the identity relation

\{\langle s, s \rangle \mid s \in S\}

.^[41] Such constructions appear in contexts like projections in product spaces, where a partial identity might embed a subspace while ignoring elements outside it. In the multivariable setting, the identity function on

\mathbb{R}^n

extends componentwise as

\mathrm{id}(x_1, \dots, x_n) = (x_1, \dots, x_n)

, preserving the vector structure and serving as the neutral element under function composition in the space of maps from

\mathbb{R}^n

to itself.^[42] This generalization maintains bijectivity and continuity, aligning with its role in linear algebra as the identity transformation represented by the

n \times n

identity matrix.

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