Showing a binary operation is well defined

Let $G=\{x\in\mathbb{R}~|~0\le x<1\}$ and define $\star$ as follows

$$x\star y=x+y-[x+y]$$

I want to show this is a well defined operation. I think it is this easy;

If $x=x', y=y'$, then

$$x\star y = x+y-[x+y]=x'+y'-[x'+y']=x'\star y'$$

Is this accurate?

1 Answer

We have:

$$\lfloor x \rfloor \le x \lt \lfloor x \rfloor + 1$$

so that:

$$0 \le x-\lfloor x \rfloor\lt 1$$

So your function is well defined $$x\star y:G\to G$$