Answer to Question #116069 in Discrete Mathematics for Joshua

Question

The binary operation on Z given by x
mathrmast (y=1+xy) not in the question later appered when i pasted)
is______ but not associative
a.Associative
b.Commutative
c.Distributive
d.Additive

Answer 1

Let $y=1+xy$ then

$xy=y-1$

The binary operation is not associative :

$(xy)z\neq x(yz)\\ (xy)z=(y-1)z=yz-z=z-1-z=-1\\ x(yz)=x(z-1)=xz-x=z-1-x$

The binary operation is not commutative: 

$xy\neq yx\\ xy=y-1\\ yx=x-1$

The binary operation is not distributive:

$(x+y)z\neq xz+yz\\ (x+y)z=z-1\\ xz+yz=xz+yz=z-1+z-1=2z-2\\ z(x+y)z\neq zx+zy\\ z(x+y)=x+y-1\\ zx+zy=x-1+y-1=x+y-2\\$

 The binary operation is additive

Answer d