Left Operation is not Commutative

Theorem
Let $S$ be a finite set.

Let $\leftarrow$ denote the left operation on $S$.

Then $\leftarrow$ is not commutative on $S$ unless $S$ is a singleton.

Proof
Let $S$ be a singleton, $S = \set s$, say.

Then:
 * $s \leftarrow s = s$

and so $\leftarrow$ is trivially commutative on $S$

Otherwise, $\exists s, t \in S$ such that $s \ne t$.

Then:
 * $s \leftarrow t = s$

but:
 * $t \leftarrow s = t$

and the result follows by definition of commutative operation.

Also see

 * Left Operation is Anticommutative


 * Right Operation is not Commutative