Difference between revisions of "Commutative property"
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An [[operation]] (especially a [[binary operation]]) is said to be '''commutative''' if the order of its arguments does not affect the value. | An [[operation]] (especially a [[binary operation]]) is said to be '''commutative''' if the order of its arguments does not affect the value. | ||
| − | For example, the operation [[addition]] is commutative on the most commonly used number systems (the [[complex number]]s and its [[subset]]s such as the [[real number]]s, [[integer]]s, etc.) because <math>a + b = b + a</math>. However, the operation of [[division]] is not commutative over these sets because usually <math>\frac ab \neq \frac ba</math>. | + | For example, the operation [[addition]] is commutative on the most commonly used number systems (the [[complex number]]s and its [[subset]]s such as the [[real number]]s, [[integer]]s, etc.) because <math>\displaystyle a + b = b + a</math>. However, the operation of [[division]] is not commutative over these sets because usually <math>\frac ab \neq \frac ba</math>. |
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| + | Formally, an operation <math>G: S \to S</math> is commutative if and only if <math>\forall a, b \in S, G(a, b) = G(b, a)</math>. | ||
An operation which is not commutative is said to be [[noncommutative]]. | An operation which is not commutative is said to be [[noncommutative]]. | ||
Commutivity is especially important in [[abstract algebra]]. The study of [[group]]s in which the group operation is commutative ([[abelian group]]s) is a very important part of [[group theory]]. | Commutivity is especially important in [[abstract algebra]]. The study of [[group]]s in which the group operation is commutative ([[abelian group]]s) is a very important part of [[group theory]]. | ||
Revision as of 13:02, 24 July 2006
An operation (especially a binary operation) is said to be commutative if the order of its arguments does not affect the value.
For example, the operation addition is commutative on the most commonly used number systems (the complex numbers and its subsets such as the real numbers, integers, etc.) because 
. However, the operation of division is not commutative over these sets because usually 
.
Formally, an operation 
is commutative if and only if 
.
An operation which is not commutative is said to be noncommutative.
Commutivity is especially important in abstract algebra. The study of groups in which the group operation is commutative (abelian groups) is a very important part of group theory.
