Let $S$ be a set and $\circ$ a binary operation on it. $\circ$ is said to be commutative if

$$a \circ b = b \circ a$$

for all $a,b \in S$

Viewing $\circ$ as a function from $S\times S$ to $S$ the commutativity of $\circ$ can be notated as $$\circ(a,b)=\circ(b,a).$$

Some common examples of commutative operations are

• addition over the integers: $m+n=m+n$ for all integers $m,n$
• multiplication over the integers: $m\cdot n=m\cdot n$ for all integers $m,n$
• addition over $n \times n$ matrices, $A+B=B+A$ for all $n\times n$ matrices $A,B$ and
• multiplication over the reals: $rs=sr$ 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, $$a-b\ne b-a.$$ For instance, $2-1=1\ne -1 = 1-2$

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 $n\ge 2$ An $n$ ary operation $f$ on a set $A$ is said to be commutative if $$f(a_1,a_2,\ldots, a_n)=f(a_{\pi(1)},a_{\pi(2)},\ldots, a_{\pi(n)})$$ for every permutation $\pi$ on $\lbrace 1,2,\ldots, n\rbrace$ and for every choice of $n$ elements $a_i$ of $A$