In wikipedia, I come across this identity for the floor function:
In my question, I am interested in $x,y>0$. \begin{equation} \lfloor x \rfloor + \lfloor y \rfloor \leq \lfloor x + y \rfloor \leq \lfloor x \rfloor + \lfloor y\rfloor + 1. \end{equation}
Now, making some substitutions I arrive at \begin{equation} \lfloor x - y \rfloor \leq \lfloor x \rfloor - \lfloor y \rfloor \leq \lfloor x - y \rfloor + 1. \end{equation}
However, I think that because of my positivity restriction, it is even stronger than that. \begin{equation} \lfloor x - y \rfloor \leq \lfloor x \rfloor - \lfloor y \rfloor < \lfloor x - y \rfloor + 1. \end{equation} Is this true?
Or is there a case where we have equality \begin{equation}\label{key} \lfloor x \rfloor - \lfloor y \rfloor = \lfloor x - y \rfloor + 1, \end{equation} I suspect this is not possible, since the only possible `edge' cases are half integers. However, equality does not hold for such cases.
If $x = 2, y = 1.5$, then $$ \lfloor x \rfloor - \lfloor y \rfloor = 2 - 1 = 0 + 1 = \lfloor x - y \rfloor + 1. $$