Textbook Problem

Let $S$ be the set of four elements given by $S=\left\{A,B,C,D\right\}$

with the following table.

$\ast$	$A$	$B$	$C$	$D$
$A$	$B$	$C$	$A$	$B$
$B$	$C$	$D$	$B$	$A$
$C$	$A$	$B$	$C$	$D$
$D$	$A$	$B$	$D$	$D$

a. Is the binary operation $\ast$ commutative? Why?

b. Determine whether there is an identity element in $S$ for $\ast$.

c. If there is an identity element, which elements have inverses?