for all .
Viewing as a function from to , the commutativity of can be notated as
Some common examples of commutative operations are
addition over the integers: for all integers
multiplication over the integers: for all integers
addition over matrices, for all matrices , and
multiplication over the reals: , for all real numbers .
A binary operation that is not commutative is said to be non-commutative. A common example of a non-commutative operation is the subtraction over the integers (or more generally the real numbers). This means that, in general,
For instance, .
Other examples of non-commutative binary operations can be found in the attachment below.
Remark. The notion of commutativity can be generalized to -ary operations, where . An -ary operation on a set is said to be commutative if
for every permutation on , and for every choice of elements of .
|Date of creation||2013-03-22 12:22:45|
|Last modified on||2013-03-22 12:22:45|
|Last modified by||CWoo (3771)|