Let be a set and a binary operation on it. is said to be commutative if

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,

Other examples of non-commutative binary operations can be found in the attachment below.