Chapter 1.4, Problem 2E

In each part following, a rule that determines a binary operation $\ast$ on the set $Z$ of all integers is given. Determine in each case whether the operation is commutative or associative and whether there is an identity element. Also find the inverse of each invertible element.

$x\ast y=x+xy$ b. $x\ast y=x$.

$x\ast y=x+2y$ d. $x\ast y=3\left(x+y\right)$.

$x\ast y=3xy$ f. $x\ast y=x-y$.

$x\ast y=x+xy+y$ h. $x\ast y=x+y+3$.

$x\ast y=x-y+1$ j. $x\ast y=x+xy+y-2$.

$x\ast y=\left|x\right|-\left|y\right|$ l. $x\ast y=\left|x-y\right|$.

$x\ast y=x^y$ for $x,y\in Z^{+}$ n. $x\ast y=2^{xy}$ for $x,y\in Z^{+}$