Just want to point out, the last example is also an example of a set with a binary operation that is associative but NOT commutative.
Let G = $\mathbb{Z}$, and let $*$ be the following operation.
$*:\mathbb{Z} \times \mathbb{Z}\rightarrow $ given by $x*y = x$
 Take $\forall x, y, z\in \mathbb{Z},
 (x*(y*z)) = x*y = x ((x*y)*z) = x*z = x$
$\therefore$ associative.
 But, $x*y = x \neq y*x = y$
Hence it is associative but NOT commutative.

i'll convert this later for anyone interested *lol* just gna dump this here first

edited:
oooo latex is working on blogger ITS BEAUTIFUL