Commutative property | EPFL Graph Search

Are you an EPFL student looking for a semester project?

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of the property that says something like "3 + 4 = 4 + 3" or "2 × 5 = 5 × 2", the property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, "3 − 5 ≠ 5 − 3"); such operations are not commutative, and so are referred to as noncommutative operations. The idea that simple operations, such as the multiplication and addition of numbers, are commutative was for many years implicitly assumed. Thus, this property was not named until the 19th century, when mathematics started to become formalized. A similar property exists for binary relations; a binary relation is said to be symmetric if the relation applies regardless of the order of its operands; for example, equality is symmetric as two equal mathematical objects are equal regardless of their order. A binary operation on a set S is called commutative if An operation that does not satisfy the above property is called non-commutative. One says that x commutes with y or that x and y commute under if In other words, an operation is commutative if every two elements commute. Addition and multiplication are commutative in most number systems, and, in particular, between natural numbers, integers, rational numbers, real numbers and complex numbers. This is also true in every field. Addition is commutative in every vector space and in every algebra. Union and intersection are commutative operations on sets. "And" and "or" are commutative logical operations. Some noncommutative binary operations: Equation xy = yx Division is noncommutative, since . Subtraction is noncommutative, since . However it is classified more precisely as anti-commutative, since . Exponentiation is noncommutative, since .

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (7)

Hopf algebras and Hopf-Galois extensions in infinity-categories

Aras Ergus

In this thesis, we study interactions between algebraic and coalgebraic structures in infinity-categories (more precisely, in the quasicategorical model of (infinity, 1)-categories). We define a notio

EPFL2022

Relative, local and global dimension in complex networks

Alexis Arnaudon

Dimension is a fundamental property of objects and the space in which they are embedded. Yet ideal notions of dimension, as in Euclidean spaces, do not always translate to physical spaces, which can b

NATURE PORTFOLIO2022

A data-science approach to predict the heat capacity of nanoporous materials

Berend Smit, Elias Moubarak, Balázs Álmos Novotny, Seyedmohamad Moosavi, Daniele Ongari, Mehrdad Asgari, Andres Adolfo Ortega Guerrero, Özge Kadioglu

The heat capacity of a material is a fundamental property of great practical importance. For example, in a carbon capture process, the heat required to regenerate a solid sorbent is directly related t

NATURE PORTFOLIO2022

Related concepts (171)

Related courses (19)

MATH-506: Topology IV.b - cohomology rings

Singular cohomology is defined by dualizing the singular chain complex for spaces. We will study its basic properties, see how it acquires a multiplicative structure and becomes a graded commutative a

CS-212: System programming project

L'objectif de ce cours à projet est de donner aux étudiants une expérience de la pratique de la programmation système : écriture, correction, amélioration et analyse critique de leur code.

MATH-101(g): Analysis I

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable.

Related lectures (195)

Binary operation

In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary operation on a set is a binary operation whose two domains and the codomain are the same set. Examples include the familiar arithmetic operations of addition, subtraction, and multiplication. Other examples are readily found in different areas of mathematics, such as vector addition, matrix multiplication, and conjugation in groups.

Commutative property

Anticommutative property

In mathematics, anticommutativity is a specific property of some non-commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the inverse of the result with unswapped arguments. The notion inverse refers to a group structure on the operation's codomain, possibly with another operation. Subtraction is an anticommutative operation because commuting the operands of a − b gives b − a = −(a − b); for example, 2 − 10 = −(10 − 2) = −8.

Covers the properties of rational numbers and introduces the concept of an ordered field in Q.

Discusses the significance of moving parentheses before calculations and the rules for simplifying expressions and prioritizing operations.

Explores properties of operations, equivalence relations, and logic basics.