20-00-Commutative.tex

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Let $S$ be a set and $\circ$ a binary operation on it.  $\circ$ is said to be \emph{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 
\begin{itemize}
\item addition over the integers: $m+n=m+n$ for all integers $m,n$
\item multiplication over the integers: $m\cdot n=m\cdot n$ for all integers $m,n$
\item addition over $n \times n$ matrices, $A+B=B+A$ for all $n\times n$ matrices $A,B$, and
\item multiplication over the reals: $rs=sr$, for all real numbers $r,s$.
\end{itemize}

A binary operation that is not commutative is said to be \emph{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.

\textbf{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 \emph{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 $A$.
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