Section 1
A
Prove that each of the following sets, with the indicated operation, is an abelian group.$x * y=x+y+k$( $k$ a fixed constant), on the set $\mathbb{R}$ of the real numbers.
Prove that each of the following sets, with the indicated operation, is an abelian group.$$x * y=\frac{x y}{2}, \text { on the set }\{x \in \mathbb{R}: x \neq 0\}$$
Prove that each of the following sets, with the indicated operation, is an abelian group.$$x * y=x+y+x y, \text { on the set }\{x \in \mathbb{R}: x \neq-1\}$$
Prove that each of the following sets, with the indicated operation, is an abelian group.$$x * y=\frac{x+y}{x y+1}, \text { on the set }\{x \in \mathbb{R} ;-1<x<1\}$$