Commutative Property

Come friends, today we will know about the commutative Property. In mathematics , a binary operation is commutative if the order of the transformed operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar with property names that say “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 , which do not have this (for example, “3 − 5 5 − 3” ); Such operations are not commutative , and are therefore called non-commutative operations ., The idea that simple operations, such as multiplication and addition of numbers , are commutative, was assumed indirectly for many years. Thus, this property was not named until the 19th century, when mathematics began to be formalised. A related property exists for binary relations ; A binary relation is said to be symmetric if the relation applies regardless of its order of operations; For example, equality is symmetric because two identical mathematical objects are identical regardless of their order.

General use

Commutativity (or commutativity law ) is a property generally associated with binary operations and functions . If the commutative property holds for a pair of elements under a certain binary operation then the two elements are said to be commutative under that operation .

Mathematical definitions

The term “commutative” is used in several related senses.

  1. A binary operation on a set S is said to be commutative if: *
x*y=y*x~For All~x,y~ \epsilon ~S
  1. An operation that does not satisfy the above property is said to be non-commutative . One says that x commutes under y if : x * y=y*x
  2. A binary function is said to be commutative if: f: A*A ->B
 f(x,y)=f(y,x)~For All x,y,~ \epsilon A


Commutative operations in everyday life

  • Wearing socks resembles a commutative operation, since the sock is put on first, this is insignificant. Either way, the result (having both socks on), is the same. In contrast, wearing underwear and trousers is not commutative.
  • Commutativity of addition is observed when paying for an item with cash. Regardless of the order in which the bills are assigned, they always give the same total.

Commutative Operations in Mathematics

Two well-known examples of commutative binary operations: 

  • The real numbers of addition are commutative, because
 y+z=z+y~For All~~  y,z~~ \epsilon ~~R

For example, 4 + 5 = 5 + 4, because both expressions are equal to 9.

  • The real numbers of multiplication are commutative, because
yz=zy~for~ all~ ~y,z ~ \epsilon~ R

For example, 3 × 5 = 5 × 3, because both expressions are equal to 15.

As a direct consequence, it is also true that the expressions in the form y% of z and z% of y are commutative for all real numbers y and z. [6] For example 64% of 50 = 50% of 64, because both expressions are equal to 32, and 30% of 50% = 50% of 30%, because both of these expressions are equal to 15%.

  • Some binary truth functions are also commutative, because when one changes the order of the operands the truth tables for the function are the same.For example, the logical two-conditional function pq is equivalent to q p. This function is also written as p IFF q, or p q, or E pq . The final form is an example of the most concise notation in the article on Truth Functions, which lists sixteen possible binary truth functions of which eight are commutative: VPQ = VQP ; A PQ (OR) = A QP ; d pq (nand) = d qp ; E PQ (IFF) = E QP ; J PQ = J QP ; k pq (and) = k qp ; X PQ (NOR) = X QP ; o pq = o qp .
  • Further examples of commutative binary operations include addition and multiplication of complex numbers, addition and scalar multiplication of vectors , and intersection and union of sets .

Non-Genetic Operations in Daily Life

  • Concatenation , the act of concatenating character strings together, is a noncommutative operation. for example, EA + T = EAT TEA = T + EA
  • Laundry and drying resembles a noninvasive operation; Washing and then drying produces a different result than drying and then washing.
  • Rotating a book 90° around a vertical axis then 90° around a horizontal axis produces a different orientation than rotating it in the opposite order.
  • Rubik’s cube twists are noncommutative. This can be studied using group theory .
  • Thought processes are non-transcendental: a person asked a question (a) and then a question (b) may answer each question differently than a person asked first (b) and then (a), because Asking a question can change a person’s state of mind.
  • The function of dressing is either commutative or non-commutative, depending on the objects. Wearing underwear and normal clothing is non-genetic. Wearing left and right socks is commutative.
  • Shuffling a deck of cards is non-indexable. Given A and B, two ways of shuffling a deck of cards, doing A first and then B in general is not the same as shuffling B first and then A.

Noncommutative Operations in Mathematics

Some noncommutative binary operations:

division and subtraction

Division is non-transcendental, because .

{\displaystyle 1\div 2\neq 2\div 1}

Subtraction is non-transcendental, because . Although it is more accurately classified as anti-commutative , since.

{\displaystyle 0-1\neq 1-0}{\displaystyle 0-1=-(1-0)}

true work

Some truth functions are noncommutative because when one changes the order of the operands the truth tables for the function are different. For example, the truth tables for (A B) = (¬A B) and (B A) = (A B) are

aba bb a

Working Structure of Linear Functions

The function structure of linear functions from real numbers to real numbers is almost always noncommutative. For example, let and . Then

{\displaystyle f(x)=2x+1}{\displaystyle ~g(x)=3x+7}
{\displaystyle (f\circ g)(x)=f(g(x))=2(3x+7)+1=6x+15}


{\displaystyle (g\circ f)(x)=g(f(x))=3(2x+1)+7=6x+10}

It is also generally applied for linear and affine transformations from a vector space to itself (see below for a matrix representation).

matrix multiplication

Matrix multiplication of square matrices is almost always noncommutative, for example:

{\displaystyle {\begin{bmatrix}0&2\\0&1\end{bmatrix}}={\begin{bmatrix}1&1\\0&1\end{bmatrix}}{\begin{bmatrix}0&1\\0&1\end{bmatrix}}\neq {\begin{bmatrix}0&1\\0&1\end{bmatrix}}{\begin{bmatrix}1&1\\0&1\end{bmatrix}}={\begin{bmatrix}0&1\\0&1\end{bmatrix}}}

vector product

The vector product (or cross product) of two vectors in three dimensions is anti-commutative; i.e., B × A = -( A × B ).

History and Etymology

Records of the implicit use of the commutative property go back to ancient times. The Egyptians used commutative multiplication products to make computing easier. [8] [9] Euclid is known to have assumed the commutative property of multiplication in his book Elements. [10] The formal use of the permutation property dates back to the late 18th and early 19th centuries, when mathematicians began to work on the theory of functions. Today the commutative property is a well-known and basic property used in most branches of mathematics.

The first recorded use of the term commutative was in a memoir by François Servois in 1814, [1] [11] in which the term commutative was used when describing functions that are now called commutative properties. The word is a combination of the French word commuter meaning “to substitute or switch” and the suffix – meaning “to tend”, so the word literally means “tend to substitute or switch”. The term was then published in English in 1838 [2] in Duncan Farquharson’s article entitled “On the Real Nature of Symbolic Algebra” published in 1840 in Gregory’s Transactions of the Royal Society of Edinburgh.

propositional logic

rule of substitution

In truth-functional propositional logic, commutation , or commutativity  refers to two valid laws of substitution. Rules allow propositional variables to be moved within logical expressions in logical proofs. The rules are:

{\displaystyle (P\lor Q)\Leftrightarrow (Q\lor P)}


{\displaystyle (P\land Q)\Leftrightarrow (Q\land P)}

Where ” ” represents a metallurgy symbol which can be replaced with “” in a proof.

true functional coordinator

The commutativity truth is a property of some logical connectives of functional propositional logic. The following logical analogies show that commutativity is a property of special combinators. The following are true-functional tautology.

Commutativity of Combination

{\displaystyle (P\land Q)\leftrightarrow (Q\land P)}

commutativity of discontinuity

{\displaystyle (P\lor Q)\leftrightarrow (Q\lor P)}

Permutation of Implications (also called the Law of Permutations)

(P\to (Q\to R))\leftrightarrow (Q\to (P\to R))

Permutation of Equivalence (also called Completely Commutative Law of Equivalence)

(P\leftrightarrow Q)\leftrightarrow (Q\leftrightarrow P)

set theory

In group and set theory, many algebraic structures are said to be commutative when certain operands satisfy commutativity. In higher branches of mathematics, such as analysis and linear algebra, the permutations of well-known operations (such as addition and multiplication on real and complex numbers) are often used (or implicitly assumed) in proofs. 

Mathematical Structures and Commutativity

  • A commutative semigroup is a set endowed with aggregate, associative and commutative operations.
  • If the operation additionally has an identity element, then we have a commutative monoid
  • An abelian group, or commutative group, is a group whose group operations are commutative.
  • An commutative ring is a ring whose multiplication is commutative. (Additions to the ring are always commutative.)
  • In a field, both addition and multiplication are commutative. 

related properties


The associative property is closely related to the commutative property. The associative property of an expression containing two or more occurrences of the same operator states that the order is performed in the operation, so long as the order of the terms does not change, the final result is not affected. In contrast, the commutative property states that the order of the terms does not affect the final result.

Most of the commutative operations in practice are also associative. However, commutativity does not imply associativity. is a copy instance task

f(x,y)={\frac {x+y}{2}},

which is clearly commutative ( exchange of x and y does not affect the result), but it is not associative (since, for example, but a ) More such examples can be found in commutative non-associative magma.




Some forms of symmetry can be directly linked to commutativity. When a commutative operator is written as a binary function, the resulting function is symmetric in the line . As an example, if we represent addition (a commutative operation) to a function f , so that then then is a symmetric function, which can be seen in the adjoining image.


For relations, a symmetric relation corresponds to a commutative operation, in that if a relation R is symmetric, then .

aRb \Leftrightarrow bRa

Non-Commuting Operators in Quantum Mechanics

In quantum mechanics as formulated by Schrödinger, physical variables are represented by linear operators such as (which means to multiply ), and . These two operators do not travel as can be seen by looking at the effect of their compositions and (also called products of operators) on the one-dimensional wave function:

xx{\textstyle {\frac {d}{dx}}}{\textstyle x{\frac {d}{dx}}}{\textstyle {\frac {d}{dx}}x}\psi (x)
{\displaystyle x\cdot {\mathrm {d} \over \mathrm {d} x}\psi =x\cdot \psi '\ \neq \ \psi +x\cdot \psi '={\mathrm {d} \over \mathrm {d} x}\left(x\cdot \psi \right)}

According to Heisenberg’s uncertainty principle, if two operators representing a pair of variables do not interact, then that pair of variables is complementary, meaning that they cannot be measured or precisely known together. For example, in position and linear motion – the direction of a particle is represented by the operators and , respectively (where is less Planck constant). This is the same example except for the constants , so again the operators do not move and the physical meaning is that position and linear momentum in a given direction are complementary.

xx-i\hbar {\frac {\partial }{\partial x}}\hbar-i\hbar
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