Commutative property

Template:Short description Template:Redirect Template:Use dmy dates Template:Infobox mathematical statement

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. Template:Nowrap or Template:Nowrap, 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, Template:Nowrap); 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.Template:Sfn

DefinitionEdit

A binary operation <math>*</math> on a set S is commutative if <math display="block">x * y = y * x </math> for all <math> x,y \in S</math>.Template:Sfn An operation that is not commutative is said to be noncommutative.Template:Sfn

One says that Template:Mvar commutes with Template:Math or that Template:Mvar and Template:Mvar commute under <math>*</math> ifTemplate:Sfn <math display="block"> x * y = y * x.</math>

So, an operation is commutative if every two elements commute.Template:Sfn 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.Template:Sfn

ExamplesEdit

File:Commutative Addition.svg

The cumulation of apples, which can be seen as an addition of natural numbers, is commutative.

Commutative operationsEdit

File:Vector Addition.svg

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 systems, and, in particular, between natural numbers, integers, rational numbers, real numbers and complex numbers. This is also true in every field.Template:Sfn
Addition is commutative in every vector space and in every algebra.Template:Sfn
Union and intersection are commutative operations on sets.Template:Sfn
"And" and "or" are commutative logical operations.Template:Sfn

Noncommutative operationsEdit

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 anti-commutative, since <math>x - y = - (y - x)</math> for every Template:Tmath and Template:Tmath. Exponentiation is noncommutative, since <math>2^3\neq3^2</math> (see Equation x^y = y^x.Template:Sfn
Some truth functions are noncommutative, since their truth tables are different when one changes the order of the operands.Template:Sfn For example, the truth tables for Template:Math and Template:Math are

Template:Aligned table

Function composition is generally noncommutative.Template:Sfn 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 Template:Tmath matrices. For example:Template:Sfn <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 anti-commutative; i.e., <math> \mathbf{b} \times \mathbf{a} = -(\mathbf{a} \times \mathbf{b}) </math>.Template:Sfn

Commutative structuresEdit

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;Template:Sfn
a commutative monoid is a monoid whose operation is commutative;Template:Sfn
a commutative group or abelian group is a group whose operation is commutative;Template:Sfn
a commutative ring is a ring whose multiplication is commutative. (Addition in a ring is always commutative.)Template:Sfn

However, in the case of algebras, the phrase "commutative algebra" refers only to associative algebras that have a commutative multiplication.Template:Sfn

History and etymologyEdit

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.Template:Sfn Euclid is known to have assumed the commutative property of multiplication in his book Elements.<ref>Template:Harvnb. See 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.Template:Sfn

File:Commutative Word Origin.PNG

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 Servois in 1814, which used the word commutatives when describing functions that have what is now called the commutative property.Template:Sfn 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.Template:Sfnm

NotesEdit

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ReferencesEdit

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