Commutative property#Definition

{{Infobox mathematical statement

| name = {{PAGENAMEBASE}}

| image = File:Commutativity of binary operations (without question mark).svg

| statement = A binary operation is commutative if changing the order of the operands does not change the result.

| symbolic statement = $x * y = y * x \quad\forall x,y\in S.$

}}

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. Perhaps most familiar as a property of arithmetic, e.g. {{nowrap|1="3 + 4 = 4 + 3"}} or {{nowrap|1="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, {{nowrap|"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 centuries implicitly assumed. Thus, this property was not named until the 19th century, when new algebraic structures started to be studied.{{sfn|Rice|2011|p=[https://books.google.com/books?id=YruifIx88AQC&pg=PA4 4]}}

Definition

A binary operation $*$ on a set S is commutative if

$x * y = y * x$

for all $x,y \in S$ .{{sfn|Saracino|2008|p=[https://books.google.com/books?id=GW4fAAAAQBAJ&pg=PA11 11]}} An operation that is not commutative is said to be noncommutative.{{sfn|Hall|1966|pp=[https://books.google.com/books?id=qqs8AAAAIAAJ&pg=PA262 262–263]}}

One says that {{mvar|x}} commutes with {{math|y}} or that {{mvar|x}} and {{mvar|y}} commute under $*$ if{{sfn|Lovett|2022|p=[http://books.google.com/books?id=vp0IEQAAQBAJ&pg=PA12 12]}}

$x * y = y * x.$

So, an operation is commutative if every two elements commute.{{sfn|Lovett|2022|p=[http://books.google.com/books?id=vp0IEQAAQBAJ&pg=PA12 12]}} An operation is noncommutative if there are two elements such that $x * y \ne y * x.$ This does not exclude the possibility that some pairs of elements commute.{{sfn|Hall|1966|pp=[https://books.google.com/books?id=qqs8AAAAIAAJ&pg=PA262 262–263]}}

Examples

File:Commutative Addition.svg

= Commutative operations =

File:Vector Addition.svg

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.{{sfn|Rosen|2013|loc = See the [https://books.google.com/books?id=-oVvEAAAQBAJ&pg=SL1-PA1 Appendix I]}}
Addition is commutative in every vector space and in every algebra.{{sfn|Sterling|2009|p=[https://books.google.com/books?id=PsNJ1alC-bsC&pg=PA248 248]}}
Union and intersection are commutative operations on sets.{{sfn|Johnson|2003|p=[http://books.google.com/books?id=-pZX2KS2KqMC&pg=PA642 642]}}
"And" and "or" are commutative logical operations.{{sfn|O'Regan|2008|p=[https://books.google.com/books?id=081H96F1enMC&pg=PA33 33]}}

= Noncommutative operations =

Division is noncommutative, since $1 \div 2 \neq 2 \div 1$ . Subtraction is noncommutative, since $0 - 1 \neq 1 - 0$ . However it is classified more precisely as anti-commutative, since $x - y = - (y - x)$ for every {{tmath|x}} and {{tmath|y}}. Exponentiation is noncommutative, since $2^3\neq3^2$ (see Equation xy = yx{{!}}Equation x^y = y^x.{{sfn|Posamentier|Farber|Germain-Williams|Paris|2013|p=[https://books.google.com/books?id=VfCgAQAAQBAJ&pg=PA71 71]}}
Some truth functions are noncommutative, since their truth tables are different when one changes the order of the operands.{{sfn|Medina|Ojeda-Aciego|Valverde|Vojtáš|2004|p=[https://books.google.com/books?id=yAH6BwAAQBAJ&pg=PA617 617]}} For example, the truth tables for {{math|(A ⇒ B) {{=}} (¬A ∨ B)}} and {{math|(B ⇒ A) {{=}} (A ∨ ¬B)}} are

: {{aligned table

| class=wikitable

| style=text-align:center; width:20%;

| cols=4

| col3style=width:30%;

| col4style=width:30%;

| row1header=yes

| {{math|A}} | {{math|B}} | {{math|A ⇒ B}} | {{math|B ⇒ A}}

| F | F | T | T

| F | T | T | F

| T | F | F | T

| T | T | T | T

}}

Function composition is generally noncommutative.{{sfn|Tarasov|2008|p=[http://books.google.com/books?id=pHK11tfdE3QC&pg=PA56 56]}} For example, if $f(x)=2x+1$ and $g(x)=3x+7$ . Then $(f \circ g)(x) = f(g(x)) = 2(3x+7)+1 = 6x+15$ and $(g \circ f)(x) = g(f(x)) = 3(2x+1)+7 = 6x+10.$
Matrix multiplication of square matrices of a given dimension is a noncommutative operation, except for {{tmath|1\times 1}} matrices. For example:{{sfn|Cooke|2014|p=[https://books.google.com/books?id=b_iJAwAAQBAJ&pg=PA7 7]}}

\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}

The vector product (or cross product) of two vectors in three dimensions is anti-commutative; i.e., $\mathbf{b} \times \mathbf{a} = -(\mathbf{a} \times \mathbf{b})$ .{{sfn|Haghighi|Kumar|Mishev|2024|p=[http://books.google.com/books?id=D2b8EAAAQBAJ&pg=PA118 118]}}

Commutative structures

Some types of algebraic structures involve an operation that does not require commutativity. If this operation is commutative for a specific structure, the structure is often said to be commutative. So,

a commutative semigroup is a semigroup whose operation is commutative;{{sfn|Grillet|2001|pp=1–2}}
a commutative monoid is a monoid whose operation is commutative;{{sfn|Grillet|2001|p=3}}
a commutative group or abelian group is a group whose operation is commutative;{{sfn|Gallian|2006|p=34}}
a commutative ring is a ring whose multiplication is commutative. (Addition in a ring is always commutative.){{sfn|Gallian|2006|p=236}}

However, in the case of algebras, the phrase "commutative algebra" refers only to associative algebras that have a commutative multiplication.{{sfn|Tuset|2025|p=[https://books.google.com/books?id=1RE1EQAAQBAJ&pg=PA99 99]}}

History and etymology

Records of the implicit use of the commutative property go back to ancient times. The Egyptians used the commutative property of multiplication to simplify computing products.{{sfn|Gay|Shute|1987|p=[https://archive.org/details/rhindmathematica0000robi_h8l4/page/16 16‐17]}} Euclid is known to have assumed the commutative property of multiplication in his book Elements.{{harvnb|Barbeau|1968|p=183}}. See [http://aleph0.clarku.edu/~djoyce/elements/bookVII/propVII5.html Book VII, Proposition 5], in David E. Joyce's online edition of Euclid's Elements Formal uses of the commutative property arose in the late 18th and early 19th centuries when mathematicians began to work on a theory of functions. Nowadays, the commutative property is a well-known and basic property used in most branches of mathematics.{{sfn|Saracino|2008|p=[https://books.google.com/books?id=GW4fAAAAQBAJ&pg=PA11 11]}}

File:Commutative Word Origin.PNG

The first recorded use of the term commutative was in a memoir by François Servois in 1814, which used the word commutatives when describing functions that have what is now called the commutative property.{{sfn|Allaire|Bradley|2002}} Commutative is the feminine form of the French adjective commutatif, which is derived from the French noun commutation and the French verb commuter, meaning "to exchange" or "to switch", a cognate of to commute. The term then appeared in English in 1838. in Duncan Gregory's article entitled "On the real nature of symbolical algebra" published in 1840 in the Transactions of the Royal Society of Edinburgh.{{sfnm

| 1a1 = Rice | 1y = 2011 | 1p = [https://books.google.com/books?id=YruifIx88AQC&pg=PA4 4]

| 2a1 = Gregory | 2y = 1840

}}

Notes

References

{{cite journal

| last1 = Allaire | first1 = Patricia R.

| last2 = Bradley | first2 = Robert E.

| year = 2002

| title = Symbolical Algebra as a Foundation for Calculus: D. F. Gregory's Contribution

| journal = Historia Mathematica

| volume = 29

| pages = 395–426

| doi = 10.1006/hmat.2002.2358

}}

{{cite book

| last = Barbeau | first = Alice Mae

| year = 1968

| title = A Historical Approach to the Theory of Groups

| volume = 2

| publisher = University of Wisconsin--Madison

}}

{{cite book

| last = Cooke | first = Richard G.

| year = 2014

| title = Infinite Matrices and Sequence Spaces

| publisher = Dover Publications

| isbn = 978-0-486-78083-2

}}

{{cite book

| last = Gallian | first = Joseph

| title = Contemporary Abstract Algebra

| edition = 6e

| year = 2006

| isbn = 0-618-51471-6

| publisher = Houghton Mifflin

}}

{{cite book

| last1 = Gay | first1 = Robins R.

| last2 = Shute | first2 = Charles C. D.

| year = 1987

| title = The Rhind Mathematical Papyrus: An Ancient Egyptian Text

| publisher = British Museum

| url = https://archive.org/details/rhindmathematica0000robi_h8l4/page/16

| url-access = registration

| isbn = 0-7141-0944-4

}}

{{cite journal

| last = Gregory | first = D. F.

| year = 1840

| title = On the real nature of symbolical algebra

| periodical = Transactions of the Royal Society of Edinburgh

| volume = 14

| pages = 208–216

| url = https://archive.org/details/transactionsofro14royal

}}

{{cite book

| last = Grillet | first = P. A.

| doi = 10.1007/978-1-4757-3389-1

| isbn = 0-7923-7067-8

| location = Dordrecht

| mr = 2017849

| publisher = Kluwer Academic Publishers

| series = Advances in Mathematics

| title = Commutative semigroups

| volume = 2

| year = 2001}}

{{cite book

| last1 = Haghighi | first1 = Aliakbar Montazer

| last2 = Kumar | first2 = Abburi Anil

| last3 = Mishev | first3 = Dimitar

| year = 2024

| title = Higher Mathematics for Science and Engineering

| publisher = Springer

| url = http://books.google.com/books?id=D2b8EAAAQBAJ

}}

{{cite book

| last = Hall | first = F. M.

| location = New York

| mr = 197233

| publisher = Cambridge University Press

| title = An Introduction to Abstract Algebra, Volume 1

| year = 1966}}

{{cite book

| last = Johnson | first = James L.

| year = 2003

| title = Probability and Statistics for Computer Science

| publisher = John Wiley & Sons

| url = http://books.google.com/books?id=-pZX2KS2KqMC

}}

{{cite book

| last = Lovett | first = Stephen

| year = 2022

| title = Abstract Algebra: A First Course

| publisher = CRC Press

| url = http://books.google.com/books?id=vp0IEQAAQBAJ

}}

{{cite conference

| last1 = Medina | first1 = Jesús

| last2 = Ojeda-Aciego | first2 = Manuel

| last3 = Valverde | first3 = Agustín

| last4 = Vojtáš | first4 = Peter

| contribution = Towards Biresiduated Multi-adjoint Logic Programming

| editor-first1 = Ricardo | editor-last1 = Conejo

| editor-first2 = Maite | editor-last2 = Urretavizcaya

| editor-first3 = José-Luis | editor-last3 = Pérez-de-la-Cruz

| title = Current Topics in Artificial Intelligence: 10th Conference of the Spanish Association for Artificial Intelligence, CAEPIA 2003, and 5th Conference on Technology Transfer, TTIA 2003, November 12-14, 2003.

| location = San Sebastian, Spain

| year = 2004

| publisher = Springer

| doi = 10.1007/b98369

}}

{{cite book

| last = O'Regan | first = Gerard

| year = 2008

| title = A brief history of computing

| publisher = Springer

| isbn = 978-1-84800-083-4

}}

{{cite book

| last = Posamentier | first = Alfred S.

| last2 = Farber | first2 = William

| last3 = Germain-Williams | first3 = Terri L.

| last4 = Paris | first4 = Elaine

| last5 = Thaller | first5 = Bernd

| last6 = Lehmann | first6 = Ingmar

| year = 2013

| title = 100 Commonly Asked Questions in Math Class

| publisher = Corwin Press

| isbn = 978-1-4522-4308-5

}}

{{cite book

| last = Rice | first = Adrian

| year = 2011

| contribution = Introduction

| title = Mathematics in Victorian Britain

| editor1-first = Raymond | editor1-last = Flood

| editor2-first = Adrian | editor2-last = Rice

| editor3-first = Robin | editor3-last = Wilson | editor3-link = Robin Wilson (mathematician)

| publisher = Oxford University Press

| isbn = 9780191627941

}}

{{cite book

| last = Rosen | first = Kenneth

| year = 2013

| title = Discrete Maths and Its Applications Global Edition

| publisher = McGraw Hill

| isbn = 978-0-07-131501-2

}}

{{cite book

| last = Saracino | first = Dan

| year = 2008

| title = Abstract Algebra: A First Course

| edition = 2nd

| publisher = Waveland Press Inc.

}}

{{cite book

| last = Sterling | first = Mary J.

| year = 2009

| title = Linear Algebra For Dummies

| publisher = John Wiley & Sons

| isbn = 978-0-470-43090-3

}}

{{cite book

| last = Tarasov | first = Vasily

| year = 2008

| edition = 1st

| volume = 7

| title = Quantum Mechanics of Non-Hamiltonian and Dissipative Systems

| url = http://books.google.com/books?id=pHK11tfdE3QC

| publisher = Elsevier

}}

{{cite book

| last = Tuset | first = Lars

| doi = 10.1007/978-3-031-74623-9

| isbn = 978-3-031-74622-2

| mr = 4886847

| publisher = Springer | location = Cham

| title = Abstract Algebra via Numbers

| year = 2025}}

Category:Properties of binary operations

Category:Elementary algebra

Category:Rules of inference

Category:Symmetry

Category:Concepts in physics

Category:Functional analysis

Commutative property#Definition

Definition

Examples

= Commutative operations =

= Noncommutative operations =

Commutative structures

History and etymology

See also

Notes

References