commutative (original) (raw)

for all a,b∈S.

Viewing ∘ as a function from S×S to S, the commutativity of ∘ can be notated as

Some common examples of commutative operations are

• •
  addition over the integers: m+n=m+n for all integers m,n
• •
  multiplication over the integers: m⋅n=m⋅n for all integers m,n
• •
  addition over n×n matrices, A+B=B+A for all n×n matrices A,B, and
• •
  multiplication over the reals: r⁢s=s⁢r, for all real numbers r,s.

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, 2-1=1≠-1=1-2.

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

Remark. The notion of commutativity can be generalized to n-ary operations, where n≥2. An n-ary operation f on a set A is said to be commutative if

for every permutation π on {1,2,…,n}, and for every choice of n elements ai of A.