Identity Element of Binary Operations

[gbean]

Homework Statement

Determine whether the operation has an identity element.
x*y = 3xy

Homework Equations

e*x = x*e = x, if this holds, e is an identity element

The Attempt at a Solution

My attempt:
x*z = z*x = 3xz, then 3xz = x <=> z = 1/3 => e = 1/3.

But the answer key in the back of the book says that this binary operation has no identity element. Am I wrong, or is the book? Thanks!

 

[╔(σ_σ)╝]

It depends on what the relation is defined on. Are x &y integers or real numbers ? If x and y are integers then it is clear that 1/3 is not.

 

[gbean]

Yes, they are integers. Why is this obvious?

 

[╔(σ_σ)╝]

The identity has to be in the group/set . 1/3 is not an integer hence, it is not in the set in which the binary operation is defined.