MathDB

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:

\text{a})\ x*(x*y)=y

for all

x,y\in\mathbb{Z}

;

\text{b})\ (x*y)*y=x

for all

x,y\in\mathbb{Z}

.
The professor claims in his book that

1.

The operation

*

is commutative:

x*y=y*x

for all

x,y\in\mathbb{Z}

2.

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?

Problem Statement