This is on my professor's notes: $$\frac {1_{0<y<x<1} (x,y)} {1_{0,1}(x)}=1_{0,x}(y).$$ I can provide more context if you want but I believe this is sufficient for my question.
I am ultimately wondering why would $x$ not need to be in $(0,1)$ in the RHS? Is there a general way to derive this?
The reason I tagged divisibility is because I want to know what happens if ${1_{0,1}(x)}=0$? Would it then be undefined?
Division by $0$ is undefined. This means that for $x\neq 0,1$ your LHS is undefined, no matter your conventions or other definitions.
Also, you cannot just define $\frac00=1$, for this will get you into trouble. For instance it would imply that $$\frac00\cdot0=1\cdot0 =0,$$ and that $$\frac00\cdot0=\frac{0\cdot0}0=\frac00=1.$$ This implies $0=1$. This is either a contradiction or a very hard constraint on the algebraic set $x$ and $y$ live in.