Question 65
An occasionally unreliable professor has devoted his last book to a certain binary operation $*$. When this operation is applied to any two integers, the result is again an integer. The operation is known to satisfy the following axioms:
a. $x *(x * y)=y$ for all $x, y \in \mathbb{Z}$;
b. $(x * y) * y=x$ for all $x, y \in \mathbb{Z}$.
The professor claims in his book that
(C1) the operation $*$ is commutative: $x * y=y * x$ for all $x, y \in \mathbb{Z}$.
(C2) the operation $*$ is associative: $(x * y) * z=x *(y * z)$ for all $x, y, z \in \mathbb{Z}$.
Which of these claims follow from the stated axioms?
a. $x *(x * y)=y$ for all $x, y \in \mathbb{Z}$;
b. $(x * y) * y=x$ for all $x, y \in \mathbb{Z}$.
The professor claims in his book that
(C1) the operation $*$ is commutative: $x * y=y * x$ for all $x, y \in \mathbb{Z}$.
(C2) the operation $*$ is associative: $(x * y) * z=x *(y * z)$ for all $x, y, z \in \mathbb{Z}$.
Which of these claims follow from the stated axioms?