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{{Other uses|Commute (disambiguation)}}
 
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.  The commutativity of simple operations, such as [[multiplication (mathematics)|multiplication]] and [[addition]] of numbers, was for many years implicitly assumed and the property was not named until the 19th century when mathematics started to become formalized. By contrast, [[division (mathematics)|division]] and [[subtraction]] are ''not'' commutative.
 
==Common uses==
The ''commutative property'' (or ''commutative law'') is a property associated with binary operations and [[Function (mathematics)|functions]].  Similarly, if the commutative property holds for a pair of elements under a certain binary operation then it is said that the two elements ''commute'' under that operation.
 
== Propositional logic ==
{{Transformation rules}}
 
=== Rule of replacement ===
In standard truth-functional propositional logic, ''commutation'',<ref>Moore and Parker</ref><ref>Copi and Cohen</ref> or ''commutativity''<ref>Hurley</ref> refer to two [[validity|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>(P \or Q) \Leftrightarrow (Q \or P)</math>
and
:<math>(P \and Q) \Leftrightarrow (Q \and 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 \and Q) \leftrightarrow (Q \and P)</math>
'''Commutativity of disjunction'''
 
:<math>(P \or Q) \leftrightarrow (Q \or P)</math>
'''Commutativity of implication''' (also called the '''Law of permutation''')
:<math>(P \to (Q \to R)) \leftrightarrow (Q \to (P \to R))</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>Axler, p.2</ref><ref name="Gallian, p.34">Gallian, p.34</ref><ref>p. 26,87</ref>
 
==Mathematical definitions==
{{Further|Symmetric function}}
The term "commutative" is used in several related senses.<ref name="Krowne, p.1">Krowne, p.1</ref><ref>Weisstein, ''Commute'', p.1</ref>
 
1. A binary operation <math>*</math> on a [[Set (mathematics)|set]] ''S'' is called ''commutative'' if:
:<math>x * y = y * x\qquad\mbox{for all }x,y\in S</math>
 
An operation that does not satisfy the above property is called '''noncommutative'''.
 
2. One says that ''x commutes'' with ''y'' under <math>*</math> if:
:<math> x * y = y * x \,</math>
 
3. A [[binary function]] <math>f \colon A \times A \to B</math> is called ''commutative'' if:
:<math>f(x, y) = f(y, x)\qquad\mbox{for all }x,y\in A</math>
 
==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>Lumpkin, p.11</ref><ref>Gay and Shute, p.?</ref> [[Euclid]] is known to have assumed the commutative property of multiplication in his book [[Euclid's Elements|''Elements'']].<ref>O'Conner and 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="ReferenceA">Cabillón and Miller, ''Commutative and Distributive''</ref><ref>O'Conner and Robertson, ''Servois''</ref> which used the word ''commutatives'' when describing functions that have what is now called the commutative property.  The word is a combination of the French word ''commuter'' meaning "to substitute or switch" and the suffix ''-ative'' meaning "tending to" so the word literally means "tending to substitute or switch."  The term then appeared in English in ''[[Philosophical Transactions of the Royal Society]]'' in 1844.<ref name="ReferenceA"/>
 
==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 doesn't 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>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(1, f(2, 3)) = 1.75</math> but <math>f(f(1, 2), 3) = 2.25</math>).
 
===Symmetry===
[[File:Symmetry Of Addition.svg|right|thumb|200px|Graph showing the symmetry of the addition function]]
{{Main|Symmetry in mathematics}}
 
Some forms of symmetry can be directly linked to commutativity.  When a commutative operator is written as a binary function then the resulting function is symmetric across the line ''y = x''.  As an example, if we let a function ''f'' represent addition (a commutative operation) so that ''f''(''x'',''y'') = ''x'' + ''y'' then ''f'' is a symmetric function, which can be seen in the image on the right.
 
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>.
 
==Examples==
 
===Commutative operations in everyday life===
 
*Putting on socks resembles a commutative operation, since which sock is put on first is unimportant. Either way, the result (having both socks on), is the same.
*The commutativity of addition is observed when paying for an item with cash.  Regardless of the order the bills are handed over in, they always give the same total.
 
===Commutative operations in mathematics===
Two well-known examples of commutative binary operations:<ref name="Krowne, p.1"/>
* The [[addition]] of [[real number]]s is commutative, since
::<math>y + z = z + y \qquad\mbox{for all }y,z\in \mathbb{R}</math>
:For example 4 + 5 = 5 + 4, since both [[Expression (mathematics)|expression]]s equal 9.
* The [[multiplication]] of [[real number]]s is commutative, since
::<math>y z = z y \qquad\mbox{for all }y,z\in \mathbb{R}</math>
:For example, 3 × 5 = 5 × 3, since both expressions equal 15.
* Some binary [[truth function]]s are also commutative, since the [[truth table]]s for the functions are the same when one changes the order of the operands.
:For example, V''pq'' = V''qp''; A''pq'' = A''qp''; D''pq'' = D''qp''; E''pq'' = E''qp''; J''pq'' = J''qp''; K''pq'' = K''qp''; X''pq'' = X''qp''; O''pq'' = O''qp''.
*Further examples of commutative binary operations include addition and multiplication of [[complex number]]s, addition and [[scalar product|scalar multiplication]] of [[vector space|vectors]], and [[intersection (set theory)|intersection]] and [[union (set theory)|union]] of [[Set (mathematics)|sets]].
 
===Noncommutative operations in everyday life===
*[[Concatenation]], the act of joining character strings together, is a noncommutative operation. For example
:<math>EA + T = EAT \neq TEA = T + EA</math>
*Washing and drying clothes resembles a noncommutative operation; washing and then drying produces a markedly different result to drying and then washing.
*Rotating a book 90° around a vertical axis then 90° around a horizontal axis produces a different orientation than when the rotations are performed in the opposite order.
*The twists of the [[Rubik's Cube]] are noncommutative. This can be studied using [[group theory]].
 
===Noncommutative operations in mathematics===
Some noncommutative binary operations:<ref>Yark, p.1</ref>
*[[Subtraction]] is noncommutative, since <math>0-1\neq 1-0</math>
*[[Division (mathematics)|Division]]  is noncommutative, since  <math>1/2\neq 2/1</math>
*Some truth functions are noncommutative, since the truth tables for the functions are different when one changes the order of the operands. 
:For example, B''pq'' = C''qp''; C''pq'' = B''qp''; F''pq'' = G''qp''; G''pq'' = F''qp''; H''pq'' = I''qp''; I''pq'' = H''qp''; L''pq'' = M''qp''; M''pq'' = L''qp''. 
*[[Matrix (mathematics)|Matrix]] multiplication is noncommutative since
:<math>
\begin{bmatrix}
0 & 2 \\
0 & 1
\end{bmatrix}
=
\begin{bmatrix}
1 & 1 \\
0 & 1
\end{bmatrix}
\cdot
\begin{bmatrix}
0 & 1 \\
0 & 1
\end{bmatrix}
\neq
\begin{bmatrix}
0 & 1 \\
0 & 1
\end{bmatrix}
\cdot
\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., ''b'' × ''a'' =  −(''a'' × ''b'').
 
==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>Gallian p.236</ref>
* In a [[field (mathematics)|field]] both addition and multiplication are commutative.<ref>Gallian p.250</ref>
 
==Non-commuting operators in quantum mechanics==
{{Main|Uncertainty principle}}
 
In [[Introduction to quantum mechanics|quantum mechanics]] as formulated by [[Erwin Schrödinger|Schrödinger]], physical variables are represented by [[linear operators]] such as ''x'' (meaning multiply by ''x''), and <math>\frac{d}{dx}</math>. These two operators do not commute as may be seen by considering the effect of their [[Function composition|compositions]] <math>x \frac{d}{dx}</math> and <math>\frac{d}{dx} x</math> (also called products of operators) on a one-dimensional [[wave function]] <math>\psi(x)</math>:
 
::<math> x{d\over dx}\psi = x\psi' \neq {d\over dx}x\psi = \psi + x\psi' </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 ''x''-direction of a particle are represented respectively by the operators <math>x</math> and <math>-i \hbar \frac{\partial}{\partial x}</math> (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}}
*[[Anticommutativity]]
*[[Binary operation]]
*[[Commutant]]
*[[Commutative diagram]]
*[[Commutative (neurophysiology)]]
*[[Commutator]]
*[[Distributivity]]
*[[Parallelogram law]]
*[[Particle statistics]] (for commutativity in [[physics]])
*[[Truth function]]
*[[Truth table]]
 
==Notes==
{{Reflist|2}}
 
==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 |ref=harv |last=Copi |first=Irving M. |last2=Cohen |first2=Carl |title=Introduction to Logic |publisher=Prentice Hall |year=2005}}
 
*{{Cite book|first=Joseph|last=Gallian|title=Contemporary Abstract Algebra, 6e|year=2006|isbn=0-618-51471-6|publisher=Houghton Mifflin|location=Boston, Mass.}}
:''Linear algebra theory.  Explains commutativity in chapter 1, uses it throughout.''
 
*{{Cite book| first=Frederick | last=Goodman | title=Algebra: Abstract and Concrete, Stressing Symmetry, 2e | publisher=Prentice Hall | year=2003 | isbn=0-13-067342-0}}
:''Abstract algebra theory.  Uses commutativity property throughout book.
 
*{{Cite book |title=A Concise Introduction to Logic 4th edition |last=Hurley |first=Patrick  |coauthors= |year=1991 |publisher=Wadsworth Publishing }}
 
===Articles===
*http://www.ethnomath.org/resources/lumpkin1997.pdf Lumpkin, B. (1997). The Mathematical Legacy Of Ancient Egypt - A Response To Robert Palter. Unpublished manuscript.
:''Article describing the mathematical ability of ancient civilizations.''
*Robins, R. Gay, and Charles C. D. Shute. 1987. ''The Rhind Mathematical Papyrus: An Ancient Egyptian Text''. London: British Museum Publications Limited. ISBN 0-7141-0944-4
:''Translation and interpretation of the [[Rhind Mathematical Papyrus]].''
 
===Online resources===
*{{springer|title=Commutativity|id=p/c023420}}
*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''
*[http://planetmath.org/?op=getuser&id=2760 Yark]. {{PlanetMath|title=Examples of non-commutative operations|urlname=ExampleOfCommutative}}, Accessed 8 August 2007
:''Examples proving some noncommutative operations''
*O'Conner, J J and Robertson, E F. [http://www-history.mcs.st-andrews.ac.uk/HistTopics/Real_numbers_1.html MacTutor history of real numbers], Accessed 8 August 2007
:''Article giving the history of the real numbers''
*Cabillón, Julio and Miller, Jeff. [http://jeff560.tripod.com/c.html Earliest Known Uses Of Mathematical Terms], Accessed 22 November 2008
:''Page covering the earliest uses of mathematical terms''
*O'Conner, J J and Robertson, E F. [http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Servois.html MacTutor biography of François Servois], Accessed 8 August 2007
:''Biography of Francois Servois, who first used the term''
 
{{Good article}}
 
[[Category:Abstract algebra]]
[[Category:Elementary algebra]]
[[Category:Mathematical relations]]
[[Category:Rules of inference]]
[[Category:Symmetry]]
[[Category:Binary operations|*Commutativity]]
[[Category:Concepts in physics]]
[[Category:Functional analysis]]
 
{{Link GA|ca}}

Revision as of 21:26, 16 January 2014

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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. The commutativity of simple operations, such as multiplication and addition of numbers, was for many years implicitly assumed and the property was not named until the 19th century when mathematics started to become formalized. By contrast, division and subtraction are not commutative.

Common uses

The commutative property (or commutative law) is a property associated with binary operations and functions. Similarly, if the commutative property holds for a pair of elements under a certain binary operation then it is said that the two elements commute under that operation.

Propositional logic

Template:Transformation rules

Rule of replacement

In standard truth-functional propositional logic, commutation,[1][2] or commutativity[3] refer to two valid rules of replacement. The rules allow one to transpose propositional variables within logical expressions in logical proofs. The rules are:

and

where "" is a metalogical symbol representing "can be replaced in a proof with."

Truth functional connectives

Commutativity is a property of some logical connectives of truth functional propositional logic. The following logical equivalences demonstrate that commutativity is a property of particular connectives. The following are truth-functional tautologies.

Commutativity of conjunction

Commutativity of disjunction

Commutativity of implication (also called the Law of permutation)

Commutativity of equivalence (also called the Complete commutative law of equivalence)

Set theory

In group and set theory, many algebraic structures are called commutative when certain operands satisfy the commutative property. In higher branches of mathematics, such as 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.[4][5][6]

Mathematical definitions

47 year-old Podiatrist Hyslop from Alert Bay, has lots of hobbies and interests that include fencing, property developers in condo new launch singapore and handball. Just had a family trip to Monasteries of Haghpat and Sanahin. The term "commutative" is used in several related senses.[7][8]

1. A binary operation on a set S is called commutative if:

An operation that does not satisfy the above property is called noncommutative.

2. One says that x commutes with y under if:

3. A binary function is called commutative if:

History and etymology

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 Egyptians used the commutative property of multiplication to simplify computing products.[9][10] Euclid is known to have assumed the commutative property of multiplication in his book Elements.[11] 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 Servois in 1814,[12][13] which used the word commutatives when describing functions that have what is now called the commutative property. The word is a combination of the French word commuter meaning "to substitute or switch" and the suffix -ative meaning "tending to" so the word literally means "tending to substitute or switch." The term then appeared in English in Philosophical Transactions of the Royal Society in 1844.[12]

Related properties

Associativity

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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 doesn't 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

which is clearly commutative (interchanging x and y does not affect the result), but it is not associative (since, for example, but ).

Symmetry

Graph showing the symmetry of the addition function

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

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

Examples

Commutative operations in everyday life

  • Putting on socks resembles a commutative operation, since which sock is put on first is unimportant. Either way, the result (having both socks on), is the same.
  • The commutativity of addition is observed when paying for an item with cash. Regardless of the order the bills are handed over in, they always give the same total.

Commutative operations in mathematics

Two well-known examples of commutative binary operations:[7]

For example 4 + 5 = 5 + 4, since both expressions equal 9.
For example, 3 × 5 = 5 × 3, since both expressions equal 15.
  • Some binary truth functions are also commutative, since the truth tables for the functions are the same when one changes the order of the operands.
For example, Vpq = Vqp; Apq = Aqp; Dpq = Dqp; Epq = Eqp; Jpq = Jqp; Kpq = Kqp; Xpq = Xqp; Opq = Oqp.

Noncommutative operations in everyday life

  • Concatenation, the act of joining character strings together, is a noncommutative operation. For example
  • Washing and drying clothes resembles a noncommutative operation; washing and then drying produces a markedly different result to drying and then washing.
  • Rotating a book 90° around a vertical axis then 90° around a horizontal axis produces a different orientation than when the rotations are performed in the opposite order.
  • The twists of the Rubik's Cube are noncommutative. This can be studied using group theory.

Noncommutative operations in mathematics

Some noncommutative binary operations:[14]

For example, Bpq = Cqp; Cpq = Bqp; Fpq = Gqp; Gpq = Fqp; Hpq = Iqp; Ipq = Hqp; Lpq = Mqp; Mpq = Lqp.
  • Matrix multiplication is noncommutative since

Mathematical structures and commutativity

Non-commuting operators in quantum mechanics

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In quantum mechanics as formulated by Schrödinger, physical variables are represented by linear operators such as x (meaning multiply by x), and . These two operators do not commute as may be seen by considering the effect of their compositions and (also called products of operators) on a one-dimensional wave function :

According to the uncertainty principle of Heisenberg, if the two operators representing a pair of variables do not commute, then that pair of variables are mutually complementary, which means they cannot be simultaneously measured or known precisely. For example, the position and the linear momentum in the x-direction of a particle are represented respectively by the operators and (where is the reduced Planck constant). This is the same example except for the constant , 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

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Notes

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References

Books

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Abstract algebra theory. Covers commutativity in that context. Uses property throughout book.
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  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

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Linear algebra theory. Explains commutativity in chapter 1, uses it throughout.
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

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Abstract algebra theory. Uses commutativity property throughout book.
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Articles

Article describing the mathematical ability of ancient civilizations.
  • Robins, R. Gay, and Charles C. D. Shute. 1987. The Rhind Mathematical Papyrus: An Ancient Egyptian Text. London: British Museum Publications Limited. ISBN 0-7141-0944-4
Translation and interpretation of the Rhind Mathematical Papyrus.

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  • Krowne, Aaron, Template:PlanetMath, Accessed 8 August 2007.
Definition of commutativity and examples of commutative operations


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Explanation of the term commute
Examples proving some noncommutative operations
Article giving the history of the real numbers
Page covering the earliest uses of mathematical terms
Biography of Francois Servois, who first used the term

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  1. ↑ Moore and Parker
  2. ↑ Copi and Cohen
  3. ↑ Hurley
  4. ↑ Axler, p.2
  5. ↑ 5.0 5.1 Gallian, p.34
  6. ↑ p. 26,87
  7. ↑ 7.0 7.1 Krowne, p.1
  8. ↑ Weisstein, Commute, p.1
  9. ↑ Lumpkin, p.11
  10. ↑ Gay and Shute, p.?
  11. ↑ O'Conner and Robertson, Real Numbers
  12. ↑ 12.0 12.1 Cabillón and Miller, Commutative and Distributive
  13. ↑ O'Conner and Robertson, Servois
  14. ↑ Yark, p.1
  15. ↑ Gallian p.236
  16. ↑ Gallian p.250
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