Prove $\displaystyle x+y-\left\lfloor x+y\right\rfloor$ is a well defined binary operation.
How do I do this?
We need to know the set of definition. Let's suppose the set is $\displaystyle \mathbb{R}$ in this case, then we have to prove that for all $\displaystyle x,y\in\mathbb{R}$ , $\displaystyle x*y=x+y-\left\lfloor x+y\right\rfloor$ exists, is unique and belongs to $\displaystyle \mathbb{R}$ .
Edited: For example, Drexel28 supposed the set of definition is $\displaystyle \mathbb{Z}$ .
The set is $\displaystyle G=\{x\in\mathbb{R}: \ x\in [0,1)\}$
I am trying to show associativity.
$\displaystyle (x*y)*z=x+y-\left\lfloor x+y\right\rfloor +z-\left\lfloor x+y-\left\lfloor x+y\right\rfloor + z\right\rfloor$
I can't see how to manipulated that into:
$\displaystyle x+y+z-\left\lfloor x+y+z-\left\lfloor y+z\right\rfloor\right\rfloor$