I know using $D_1=\{(x,y)|x<y\}$, $D_2=\{(x,y)|x=y\}$, $D_3=\{(x,y)|y<x\}$, in $(\mathbb{R};<)$, $2^3$ binary relations are definable.
But how do I know that any other binary relations are undefinable?
professor gave me hint; using automorphism, if $D$ is definable in $\mathbb{R}\times\mathbb{R}$, then for $i=1,2,3,$ $D\cap D_i \neq \emptyset$ iff $D_i \subseteq D$.
I don't get any ideas how to prove this hint(but I know using hint, any other binary relations are not definable)