Commutative property
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An operation (especially a binary operation) is said to have the commutative property or 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.
Examples
- The integers commte under both addition and multiplication, but not subtraction or division.
- Some functions commute under composition. For example, the functions
, 
for all 
(with 
taking values in the positive integers) commute: 
.
