View source for 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 = Commutativity of binary operations (without question mark).svg
| 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 years implicitly assumed. Thus, this property was not named until the 19th century, when mathematics started to become formalized.<ref name="Cabillón" /><ref name=Flood11 /> A similar property exists for [[binary relation]]s; a binary relation is said to be [[symmetric relation|symmetric]] if the relation applies regardless of the order of its operands; for example, [[Equality (mathematics)|equality]] is symmetric as two equal mathematical objects are equal regardless of their order.<ref>{{MathWorld|id=SymmetricRelation|title=Symmetric Relation}}</ref>

== Mathematical definitions ==
A [[binary operation]] <math>*</math> on a [[Set (mathematics)|set]] ''S'' is called ''commutative'' if<ref name="Krowne, p.1">Krowne, p. 1</ref><ref>Weisstein, ''Commute'', p. 1</ref>
<math display="block">x * y = y * x\qquad\mbox{for all }x,y\in S.</math> 
In other words, an operation is commutative if every two elements commute.  An operation that does not satisfy the above property is called ''noncommutative''.

One says that {{mvar|x}} ''commutes'' with {{math|''y''}} or that {{mvar|x}} and {{mvar|y}} ''commute'' under <math>*</math> if
<math display="block"> x * y = y * x.</math>
That is, a specific pair of elements may commute even if the operation is (strictly) noncommutative.

== 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]].
* Addition is commutative in every [[vector space]] and in every [[algebra over a field|algebra]].
* [[Union (set theory)|Union]] and [[intersection (set theory)|intersection]] are commutative operations on [[set (mathematics)|set]]s.
* "[[And (logic)|And]]" and "[[or (logic)|or]]" are commutative [[logical operation]]s.

=== Noncommutative operations ===
==== Division, subtraction, and exponentiation ====
{{See also|1=Equation xy = yx}}
[[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>0 - 1 = - (1 - 0)</math>.

[[Exponentiation]] is noncommutative, since <math>2^3\neq3^2</math>. This property leads to two different "inverse" operations of exponentiation (namely, the [[Nth root|''n''th-root]] operation and the [[logarithm]] operation), whereas multiplication only has one inverse operation.<ref>{{Cite web |title=User MathematicalOrchid |url=https://math.stackexchange.com/users/29949/mathematicalorchid |access-date=2024-01-20 |website=Mathematics Stack Exchange |language=en}}</ref>

==== Truth functions ====

Some [[truth function]]s are noncommutative, since the [[truth table]]s for the functions are different when one changes the order of the operands. 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 of linear functions ====
[[Function composition]] of [[linear function]]s from the [[real numbers]] to the real numbers is almost always noncommutative. For example, let <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>
This also applies more generally for [[linear map|linear]] and [[affine transformation]]s from a [[vector space]] to itself (see below for the Matrix representation).

==== Matrix multiplication ====

[[Matrix multiplication]] of [[square matrices]] is almost always noncommutative, for example:
<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>

==== Vector product ====

The vector product (or [[cross product]]) of two vectors in three dimensions is [[anticommutativity|anti-commutative]]; i.e., ''b'' × ''a'' =  −(''a'' × ''b'').

== History and etymology ==
[[File:Commutative Word Origin.PNG|right|thumb|250px|The first known use of the term was in a French Journal published in 1814]]

Records of the implicit use of the commutative property go back to ancient times. The [[Egypt]]ians used the commutative property of [[multiplication]] to simplify computing [[Product (mathematics)|products]].<ref>{{harvnb|Lumpkin|1997|p=11}}</ref><ref>{{harvnb|Gay|Shute|1987}}</ref> [[Euclid]] is known to have assumed the commutative property of multiplication in his book [[Euclid's Elements|''Elements'']].<ref>O'Conner & Robertson ''Real Numbers''</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. 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-Joseph Servois|François Servois]] in 1814,<ref name="Cabillón">{{harvnb|Cabillón|Miller|loc=''Commutative and Distributive''}}</ref><ref>O'Conner & Robertson, ''Servois''</ref> which used the word ''commutatives'' when describing functions that have what is now called the commutative property. ''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.<ref name=Flood11>{{cite book|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]]|year=2011|url=https://books.google.com/books?id=YruifIx88AQC&pg=PA4|page=4|isbn=9780191627941}}</ref> 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]].<ref>{{Cite journal|first=D. F. |last=Gregory|title=On the real nature of symbolical algebra|periodical=Transactions of the Royal Society of Edinburgh|volume=14|pages=208–216|year=1840|url=https://archive.org/details/transactionsofro14royal}}</ref>

== Propositional logic ==
{{Transformation rules}}

=== Rule of replacement ===
In truth-functional propositional logic, ''commutation'',<ref>Moore and Parker</ref><ref>{{harvnb|Copi|Cohen|2005}}</ref> or ''commutativity''<ref>{{harvnb|Hurley|Watson|2016}}</ref> refer to two [[Validity (logic)|valid]] [[rule of replacement|rules of replacement]]. The rules allow one to transpose [[propositional variable]]s within [[well-formed formula|logical expressions]] in [[formal proof|logical proofs]]. The rules are:
<math display="block">(P \lor Q) \Leftrightarrow (Q \lor P)</math>
and 
<math display="block">(P \land Q) \Leftrightarrow (Q \land P)</math>
where "<math>\Leftrightarrow</math>" is a [[metalogic]]al [[Symbol (formal)|symbol]] representing "can be replaced in a [[Formal proof|proof]] with".

=== Truth functional connectives ===

''Commutativity'' is a property of some [[logical connective]]s of truth functional [[propositional logic]]. The following [[logical equivalence]]s demonstrate that commutativity is a property of particular connectives. The following are truth-functional [[tautology (logic)|tautologies]].

;Commutativity of conjunction: <math>(P \land Q) \leftrightarrow (Q \land P)</math>
;Commutativity of disjunction: <math>(P \lor Q) \leftrightarrow (Q \lor P)</math>
;Commutativity of implication (also called the law of permutation): <math>\big(P \to (Q \to R)\big) \leftrightarrow \big(Q \to (P \to R)\big)</math>
;Commutativity of equivalence (also called the complete commutative law of equivalence): <math>(P \leftrightarrow Q) \leftrightarrow (Q \leftrightarrow P)</math>

== Set theory ==
In [[group theory|group]] and [[set theory]], many algebraic structures are called commutative when certain operands satisfy the commutative property.  In higher branches of mathematics, such as [[Mathematical analysis|analysis]] and [[linear algebra]] the commutativity of well-known operations (such as [[addition]] and [[multiplication]] on real and complex numbers) is often used (or implicitly assumed) in proofs.<ref>{{harvnb|Axler|1997|p=2}}</ref><ref name="Gallian, p. 34">{{harvnb|Gallian|2006|p=34}}</ref><ref>{{harvnb|Gallian|2006|pp=26,87}}</ref>

== Mathematical structures and commutativity ==
* A [[commutative semigroup]] is a set endowed with a total, [[associativity|associative]] and commutative operation.
* If the operation additionally has an [[identity element]], we have a [[commutative monoid]]
* An [[abelian group]], or ''commutative group'' is a [[group (mathematics)|group]] whose group operation is commutative.<ref name="Gallian, p. 34"/>
* A [[commutative ring]] is a [[ring (mathematics)|ring]] whose [[multiplication]] is commutative. (Addition in a ring is always commutative.)<ref>{{harvnb|Gallian|2006|p=236}}</ref>
* In a [[field (mathematics)|field]] both addition and multiplication are commutative.<ref>{{harvnb|Gallian|2006|p=250}}</ref>

== Related properties ==

=== Associativity ===
{{Main|Associative property}}

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 operations are performed in does not affect the final result, as long as the order of terms does not change. In contrast, the commutative property states that the order of the terms does not affect the final result.

Most commutative operations encountered in practice are also associative. However, commutativity does not imply associativity. A counterexample is the function
<math display="block">f(x, y) = \frac{x + y}{2},</math>
which is clearly commutative (interchanging ''x'' and ''y'' does not affect the result), but it is not associative (since, for example, <math>f(-4, f(0, +4)) = -1</math> but <math>f(f(-4, 0), +4) = +1</math>). More such examples may be found in [[commutative non-associative magmas]]. Furthermore, associativity does not imply commutativity either – for example multiplication of [[Quaternion#Algebraic properties|quaternions]] or of [[Matrix multiplication|matrices]] is always associative but not always commutative.

=== Distributivity ===
{{Main|Distributive property}}

=== Symmetry ===
[[File:Symmetry Of Addition.svg|right|thumb|200px|Graph showing the symmetry of the addition function]]
Some forms of [[Symmetry in mathematics|symmetry]] can be directly linked to commutativity.  When a commutative operation is written as a [[binary function]] <math>z=f(x,y),</math> then this function is called a [[symmetric function]], and its [[graph of a function|graph]] in [[three-dimensional space]] is symmetric across the plane <math>y=x</math>. For example, if the function {{math|''f''}} is defined as <math>f(x,y)=x+y</math> then <math>f</math> is a symmetric function.

For relations, a [[symmetric relation]] is analogous to a commutative operation, in that if a relation ''R'' is symmetric, then <math>a R b \Leftrightarrow b R a</math>.

== Non-commuting operators in quantum mechanics ==
{{Main|Canonical commutation relation}}

In [[Introduction to quantum mechanics|quantum mechanics]] as formulated by [[Erwin Schrödinger|Schrödinger]], physical variables are represented by [[linear operators]] such as <math>x</math> (meaning multiply by <math>x</math>), and <math display="inline">\frac{d}{dx}</math>. These two operators do not commute as may be seen by considering the effect of their [[Function composition|compositions]] <math display="inline">x \frac{d}{dx}</math> and <math display="inline">\frac{d}{dx} x</math> (also called products of operators) on a one-dimensional [[wave function]] <math>\psi(x)</math>:
<math display="block"> 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) </math>

According to the [[uncertainty principle]] of [[Werner Heisenberg|Heisenberg]], if the two operators representing a pair of variables do not commute, then that pair of variables are mutually [[complementarity (physics)|complementary]], which means they cannot be simultaneously measured or known precisely. For example, the position and the linear [[momentum]] in the <math>x</math>-direction of a particle are represented by the operators <math>x</math> and <math>-i \hbar \frac{\partial}{\partial x}</math>, respectively (where <math>\hbar</math> is the [[Planck constant|reduced Planck constant]]). This is the same example except for the constant <math>-i \hbar</math>, so again the operators do not commute and the physical meaning is that the position and linear momentum in a given direction are complementary.

== See also ==
{{Wiktionary}}
* [[Anticommutative property]]
* [[Centralizer|Centralizer and normalizer]] (also called a commutant)
* [[Commutative diagram]]
* [[Commutative (neurophysiology)]]
* [[Commutator]]
* [[Parallelogram law]]
* [[Particle statistics]] (for commutativity in [[physics]])
* [[Proofs involving the addition of natural numbers#Proof of commutativity|Proof that Peano's axioms imply the commutativity of the addition of natural numbers]]
* [[Quasi-commutative property]]
* [[Trace monoid]]
* [[Commuting probability]]

== Notes ==
{{reflist}}

== References ==

=== Books ===
* {{Cite book| first=Sheldon | last=Axler | title=Linear Algebra Done Right, 2e | publisher=Springer | year=1997 | isbn=0-387-98258-2}}
*: ''Abstract algebra theory.  Covers commutativity in that context.  Uses property throughout book.''
* {{Cite book |last1=Copi |first1=Irving M. |last2=Cohen |first2=Carl |url=https://archive.org/details/isbn_9780536179364 |title=Introduction to Logic |publisher=Prentice Hall |year=2005 |edition=12th |url-access=registration |isbn=9780131898349}}
* {{Cite book|first=Joseph|last=Gallian|title=Contemporary Abstract Algebra |edition=6e|year=2006|isbn=0-618-51471-6|publisher=Houghton Mifflin }}
*: ''Linear algebra theory.  Explains commutativity in chapter 1, uses it throughout.''
* {{Cite book| first=Frederick | last=Goodman | title=Algebra: Abstract and Concrete, Stressing Symmetry |edition=2e | publisher=Prentice Hall | year=2003 | isbn=0-13-067342-0}}
*: ''Abstract algebra theory.  Uses commutativity property throughout book.''
* {{cite book |first1=Patrick J. |last1=Hurley |first2=Lori |last2=Watson |title=A Concise Introduction to Logic |url=https://books.google.com/books?id=l-W5DQAAQBAJ&pg=PA675 |date=2016 |publisher=Cengage Learning |isbn=978-1-337-51478-1 |edition=12th}}

=== Articles ===
* {{cite web |url=http://www.ethnomath.org/resources/lumpkin1997.pdf |last=Lumpkin |first=B. |year=1997 |title=The Mathematical Legacy of Ancient Egypt – A Response To Robert Palter |archive-url=https://web.archive.org/web/20070713072942/http://www.ethnomath.org/resources/lumpkin1997.pdf |archive-date=13 July 2007 |type=Unpublished manuscript}}
*: ''Article describing the mathematical ability of ancient civilizations.''
* {{cite book |first1=Robins R. |last1=Gay |authorlink1=Gay Robins |first2=Charles C. D. |last2=Shute |authorlink2=Charles Shute (academic) |year=1987 |title=The Rhind Mathematical Papyrus: An Ancient Egyptian Text |publisher=British Museum |isbn=0-7141-0944-4}}
*: ''Translation and interpretation of the [[Rhind Mathematical Papyrus]].''

=== Online resources ===
* {{springer|title=Commutativity|id=p/c023420|ref=none}}
* Krowne, Aaron, {{PlanetMath|title=Commutative|urlname=Commutative}}, Accessed 8 August 2007.
*: ''Definition of commutativity and examples of commutative operations''
* {{MathWorld|title=Commute|urlname=Commute}}, Accessed 8 August 2007.
*: ''Explanation of the term commute''
* {{cite web |url=http://planetmath.org/?op=getuser&id=2760 |title=Yark |ref={{harvid|Yark}}}} {{PlanetMath|title=Examples of non-commutative operations|urlname=ExampleOfCommutative}}, Accessed 8 August 2007
*: ''Examples proving some noncommutative operations''
* {{cite web |last1=O'Conner |first1=J.J. |last2=Robertson |first2=E.F. |url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Real_numbers_1.html |title=History of real numbers |work=MacTutor |access-date=8 August 2007 }}
*: ''Article giving the history of the real numbers''
* {{cite web |last1=Cabillón |first1=Julio |last2=Miller |first2=Jeff |url=http://jeff560.tripod.com/c.html |title=Earliest Known Uses of Mathematical Terms |access-date=22 November 2008}}
*: ''Page covering the earliest uses of mathematical terms''
* {{cite web |last1=O'Conner |first1=J.J. |last2=Robertson |first2=E.F. |url=http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Servois.html |title=biography of François Servois |work=MacTutor |access-date=8 August 2007 |archive-date=2 September 2009 |archive-url=https://web.archive.org/web/20090902063524/http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Servois.html |url-status=dead }}
*: ''Biography of Francois Servois, who first used the term''

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[[Category:Properties of binary operations]]
[[Category:Elementary algebra]]
[[Category:Rules of inference]]
[[Category:Symmetry]]
[[Category:Concepts in physics]]
[[Category:Functional analysis]]