Actually, now that I think about, isn't

\[ \cap_X(x,x') =\begin{cases}\mathrm{true} & \mathrm{if} \ x \le x' \\\ \mathrm{false} & \mathrm{otherwise} \end{cases}\]

a bit redundant? \\([x \leq x']\\) is already *the* relation that gives \\(\mathrm{true}\\) when \\(x\\) is less then or equal to \\(x'\\), and \\(\mathrm{false}\\) otherwise. Running a conditional on \\([x \leq x']\\) to give either \\(\mathrm{true}\\) or \\(\mathrm{false}\\) is therefor redundant.

Or to be blunter,

\\[

\cap_X(x,x') := [x \leq x'].

\\]

\[ \cap_X(x,x') =\begin{cases}\mathrm{true} & \mathrm{if} \ x \le x' \\\ \mathrm{false} & \mathrm{otherwise} \end{cases}\]

a bit redundant? \\([x \leq x']\\) is already *the* relation that gives \\(\mathrm{true}\\) when \\(x\\) is less then or equal to \\(x'\\), and \\(\mathrm{false}\\) otherwise. Running a conditional on \\([x \leq x']\\) to give either \\(\mathrm{true}\\) or \\(\mathrm{false}\\) is therefor redundant.

Or to be blunter,

\\[

\cap_X(x,x') := [x \leq x'].

\\]