Editing Commutative property

{{Short description|Property of some mathematical operations}}
{{redirect|Commutative|other uses}}
{{Use dmy dates|date=June 2023}}
{{Infobox mathematical statement
| name = {{PAGENAMEBASE}}
| image = [[File:Commutativity of binary operations (without question mark).svg|220px|class=skin-invert-image]]
| type = [[Property (mathematics)|Property]]
| field = [[Algebra]]
| statement = A [[binary operation]] is ''commutative'' if changing the order of the [[operand]]s does not change the result.
| symbolic statement = <math>x * y = y * x \quad\forall x,y\in S.</math>
}}

In [[mathematics]], a [[binary operation]] is '''commutative''' if changing the order of the [[operand]]s does not change the result. It is a fundamental property of many binary operations, and many [[mathematical proof]]s 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 (mathematics)|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 (mathematics)|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 structure]]s started to be studied.{{sfn|Rice|2011|p=[https://books.google.com/books?id=YruifIx88AQC&pg=PA4 4]}}

== Definition ==

A [[binary operation]] <math>*</math> on a [[Set (mathematics)|set]] ''S'' is ''commutative'' if
<math display="block">x * y = y * x </math>
for all <math> x,y \in S</math>.{{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 <math>*</math> if{{sfn|Lovett|2022|p=[http://books.google.com/books?id=vp0IEQAAQBAJ&pg=PA12 12]}}
<math display="block"> x * y = y * x.</math>

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 <math> x * y \ne y * x.</math> 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|thumb|The cumulation of apples, which can be seen as an addition of natural numbers, is commutative.]]

=== Commutative operations ===
[[File:Vector Addition.svg|thumb|The addition of vectors is commutative, because <math>\vec a+\vec b=\vec b+ \vec a.</math>]]
* [[Addition]] and [[multiplication]] are commutative in most [[number system]]s, and, in particular, between [[natural number]]s, [[integer]]s, [[rational number]]s, [[real number]]s and [[complex number]]s. This is also true in every [[field (mathematics)|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 over a field|algebra]].{{sfn|Sterling|2009|p=[https://books.google.com/books?id=PsNJ1alC-bsC&pg=PA248 248]}}
* [[Union (set theory)|Union]] and [[intersection (set theory)|intersection]] are commutative operations on [[set (mathematics)|set]]s.{{sfn|Johnson|2003|p=[http://books.google.com/books?id=-pZX2KS2KqMC&pg=PA642 642]}}
* "[[And (logic)|And]]" and "[[or (logic)|or]]" are commutative [[logical operation]]s.{{sfn|O'Regan|2008|p=[https://books.google.com/books?id=081H96F1enMC&pg=PA33 33]}}

=== Noncommutative operations ===
* [[Division (mathematics)|Division]] is noncommutative, since <math>1 \div 2 \neq 2 \div 1</math>. [[Subtraction]] is noncommutative, since <math>0 - 1 \neq 1 - 0</math>. However it is classified more precisely as [[Anticommutativity|anti-commutative]], since <math>x - y = - (y - x)</math> for every {{tmath|x}} and {{tmath|y}}. [[Exponentiation]] is noncommutative, since <math>2^3\neq3^2</math> (see [[Equation xy = yx{{!}}Equation ''x<sup>y</sup>'' = ''y<sup>x</sup>'']].{{sfn|Posamentier|Farber|Germain-Williams|Paris|2013|p=[https://books.google.com/books?id=VfCgAQAAQBAJ&pg=PA71 71]}}
* Some [[truth function]]s are noncommutative, since their [[truth table]]s 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 <math>f(x)=2x+1</math> and <math>g(x)=3x+7</math>. Then <math display="block">(f \circ g)(x) = f(g(x)) = 2(3x+7)+1 = 6x+15</math> and <math display="block">(g \circ f)(x) = g(f(x)) = 3(2x+1)+7 = 6x+10.</math>
* [[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]}} <math display="block">
  \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}
</math>
* The vector product (or [[cross product]]) of two vectors in three dimensions is [[anticommutativity|anti-commutative]]; i.e., <math> \mathbf{b} \times \mathbf{a} = -(\mathbf{a} \times \mathbf{b}) </math>.{{sfn|Haghighi|Kumar|Mishev|2024|p=[http://books.google.com/books?id=D2b8EAAAQBAJ&pg=PA118 118]}}

==Commutative structures==
Some types of [[algebraic structure]]s 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 (mathematics)|group]] whose operation is commutative;{{sfn|Gallian|2006|p=34}}
* a [[commutative ring]] is a [[ring (mathematics)|ring]] whose [[multiplication]] is commutative. (Addition in a ring is always commutative.){{sfn|Gallian|2006|p=236}}

However, in the case of [[algebra over a field|algebras]], the phrase "[[commutative algebra (structure)|commutative algebra]]" refers only to [[associative algebra]]s 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 [[Egypt|Egyptians]] used the commutative property of [[multiplication]] to simplify computing [[Product (mathematics)|products]].{{sfn|Gay|Shute|1987|p=[https://archive.org/details/rhindmathematica0000robi_h8l4/page/16 16&dash;17]}} [[Euclid]] is known to have assumed the commutative property of multiplication in his book [[Euclid's Elements|''Elements'']].<ref>{{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''</ref> 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|right|thumb|250px|The first known use of the term was in a French Journal published in 1814]]
The first recorded use of the term ''commutative'' was in a memoir by [[François-Joseph Servois|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 [[Royal Society of Edinburgh|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
}}
{{-}}

== See also ==
{{Wiktionary}}
* [[Anticommutative property]]
* [[Canonical commutation relation]] (in quantum mechanics)
* [[Centralizer|Centralizer and normalizer]] (also called a commutant)
* [[Commutative diagram]]
* [[Commutative (neurophysiology)]]
* [[Commutator]]
* [[Particle statistics]] (for commutativity in [[physics]])
* [[Quasi-commutative property]]
* [[Trace monoid]]
* [[Commuting probability]]

== Notes ==
{{reflist}}

== References ==
{{refbegin|30em}}
* {{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
 | issue = 4
 | 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
 | isbn = 978-981-99-5431-5
 | url = https://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]]
 | isbn = 978-0-471-32672-4
 | url = https://books.google.com/books?id=-pZX2KS2KqMC
}}
* {{cite book
 | last = Lovett | first = Stephen
 | year = 2022
 | title = Abstract Algebra: A First Course
 | publisher = CRC Press
 | isbn = 978-1-000-60544-0
 | url = https://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.
 | series = Lecture Notes in Computer Science
 | location = San Sebastian, Spain
 | year = 2004
 | volume = 3040
 | publisher = Springer
 | doi = 10.1007/b98369
| isbn = 978-3-540-22218-7
 }}
* {{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
 | last1 = Posamentier | first1 = 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 = https://books.google.com/books?id=pHK11tfdE3QC
 | publisher = Elsevier
| isbn = 978-0-08-055971-1
 }}
*{{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}}
{{refend}}

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