The function <math display="inline">f:\R^2 \to \R^2</math> with <math display="inline">f(x, y) = (2x, y)</math> is a linear map. This function scales the <math display="inline">x</math> component of a vector by the factor <math display="inline">2</math>.
Injective functions. Diagramatic interpretation in the Cartesian plane, defined by the mapping
A bijective function, f: X → Y, where set X is {1, 2, 3, 4} and set Y is {A, B, C, D}. For example, f(1) = D.
3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)
The composition of two injective functions is injective.
A bijection composed of an injection (left) and a surjection (right).

In mathematical terms, a bijective function f: X → Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y.

- Bijection

Category theory is another formalization that includes also other mathematical structures and functions between structures of the same type (homomorphisms).

- Algebraic structure

A homomorphism is a map between two algebraic structures of the same type (that is of the same name), that preserves the operations of the structures.

- Homomorphism

The term must not be confused with 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.

- Injective function

An isomorphism between algebraic structures of the same type is commonly defined as a bijective homomorphism.

- Homomorphism

A homomorphism between algebraic structures is a function that is compatible with the operations of the structures.

- Injective function

For algebraic structures, monomorphisms are commonly defined as injective homomorphisms.

- Homomorphism

Arithmetics are pointed unary systems, whose unary operation is injective successor, and with distinguished element 0.

- Algebraic structure

For example, in the category Grp of groups, the morphisms must be homomorphisms since they must preserve the group structure, so the isomorphisms are group isomorphisms which are bijective homomorphisms.

- Bijection

500 related topics

Relevance

Set (mathematics)

Mathematical model for a collection of different things; a set contains elements or members, which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets.

A set of polygons in an Euler diagram
Passage with a translation of the original set definition of Georg Cantor. The German word Menge for set is translated with aggregate here.
A is a subset of B. B is a superset of A.
The intersection of A and B, denoted A ∩ B.
The relative complement
of B in A
The complement of A in U
The symmetric difference of A and B
The inclusion-exclusion principle is used to calculate the size of the union of sets: the size of the union is the size of the two sets, minus the size of their intersection.

injective (or one-to-one) if it maps any two different elements of A to different elements of B,

bijective (or a one-to-one correspondence) if the function is both injective and surjective — in this case, each element of A is paired with a unique element of B, and each element of B is paired with a unique element of A, so that there are no unpaired elements.

For example, structures in abstract algebra, such as groups, fields and rings, are sets closed under one or more operations.

Module (mathematics)

Generalization of the notion of vector space, wherein the field of scalars is replaced by a ring.

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)

If R is a ring, we can define the opposite ring Rop which has the same underlying set and the same addition operation, but the opposite multiplication: if ab = c in R, then ba = c in Rop. Any left R-module M can then be seen to be a right module over Rop, and any right module over R can be considered a left module over Rop.

This, like any homomorphism of mathematical objects, is just a mapping which preserves the structure of the objects.

A bijective module homomorphism f : M → N is called a module isomorphism, and the two modules M and N are called isomorphic.

A representation is called faithful if and only if the map R → EndZ(M) is injective.

Category theory

General theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of 20th century in their foundational work on algebraic topology.

Schematic representation of a category with objects X, Y, Z and morphisms f, g, g ∘ f. (The category's three identity morphisms 1X, 1Y and 1Z, if explicitly represented, would appear as three arrows, from the letters X, Y, and Z to themselves, respectively.)
Commutative diagram defining natural transformations

Category theory is also, in some sense, a continuation of the work of Emmy Noether (one of Mac Lane's teachers) in formalizing abstract processes; Noether realized that understanding a type of mathematical structure requires understanding the processes that preserve that structure (homomorphisms).

Category theory may be viewed as an extension of universal algebra, as the latter studies algebraic structures, and the former applies to any kind of mathematical structure and studies also the relationships between structures of different nature.

Field (mathematics)

Set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do.

The regular heptagon cannot be constructed using only a straightedge and compass construction; this can be proven using the field of constructible numbers.
The multiplication of complex numbers can be visualized geometrically by rotations and scalings.
Each bounded real set has a least upper bound.
The sum of three points P, Q, and R on an elliptic curve E (red) is zero if there is a line (blue) passing through these points.
A compact Riemann surface of genus two (two handles). The genus can be read off the field of meromorphic functions on the surface.
The fifth roots of unity form a regular pentagon.

A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.

. All field homomorphisms are injective.

(i.e., the bijections

Inverse element

Inverse element generalises the concepts of opposite (

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)

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 only if it is surjective.

An invertible homomorphism is called an isomorphism.

In this monoid, the invertible elements are the bijective functions; the elements that have left inverses are the injective functions, and those that have right inverses are the surjective functions.

A ring is an algebraic structure with two operations, addition and multiplication, which are denoted as the usual operations on numbers.

Monoid

Set equipped with an associative binary operation and an identity element.

Algebraic structures between magmas and groups. For example, monoids are semigroups with identity.

Such algebraic structures occur in several branches of mathematics.

A homomorphism between two monoids

A bijective monoid homomorphism is called a monoid isomorphism.

Mathematics

Area of knowledge that includes such topics as numbers , formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis).

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)
The distribution of prime numbers is a central point of study in number theory. This Ulam spiral serves to illustrate it, hinting, in particular, at the conditional independence between being prime and being a value of certain quadratic polynomials.
The quadratic formula expresses concisely the solutions of all quadratic equations
Rubik's cube: the study of its possible moves is a concrete application of group theory
The Babylonian mathematical tablet Plimpton 322, dated to 1800 BC.
Archimedes used the method of exhaustion, depicted here, to approximate the value of pi.
The numerals used in the Bakhshali manuscript, dated between the 2nd century BC and the 2nd century AD.
A page from al-Khwārizmī's Algebra
Leonardo Fibonacci, the Italian mathematician who introduced the Hindu–Arabic numeral system invented between the 1st and 4th centuries by Indian mathematicians, to the Western World.
Leonhard Euler created and popularized much of the mathematical notation used today.
Carl Friedrich Gauss, known as the prince of mathematicians
The front side of the Fields Medal
Euler's identity, which American physicist Richard Feynman once called "the most remarkable formula in mathematics".

The concept of algebraic structure addresses this, consisting of a set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow.

Vector space

[[File: Vector add scale.svg|200px|thumb|right|Vector addition and scalar multiplication: a vector

Addition of functions: The sum of the sine and the exponential function is
A typical matrix
Commutative diagram depicting the universal property of the tensor product.
The succeeding snapshots show summation of 1 to 5 terms in approximating a periodic function (blue) by finite sum of sine functions (red).
An affine plane (light blue) in R3. It is a two-dimensional subspace shifted by a vector x (red).

The one-to-one correspondence between vectors and their coordinates vectors maps vector addition to vector addition and scalar multiplication to scalar multiplication.

is both one-to-one (injective) and onto (surjective).

Ring (mathematics)

The integers, along with the two operations of addition and multiplication, form the prototypical example of a ring.
Richard Dedekind, one of the founders of ring theory.

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist.

Any bijective ring homomorphism is a ring isomorphism.

Category of sets

Category whose objects are sets.

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)

The epimorphisms in Set are the surjective maps, the monomorphisms are the injective maps, and the isomorphisms are the bijective maps.