intervals

how to determine if two intervals intersect

Say we have two intervals $I_1$ and $I_2$ such that $I_1=[s_1,e_1]$ and $I_2=[s_2,e_2]$.

It’s hard to enumerate all of the intersection conditions, but only two conditions satisfy the non-overlap conditions, which are the two cases where the two intervals are disjoint.

$$(e_1<s_2)\vee (e_2<s_1)$$

We can simply negate this logical statement to find a simplified expression for intersection.

$$\begin{array}{rl}& =\neg ((e_1<s_2)\vee (e_2<s_1))\\ & =\neg (e_1<s_2)\wedge \neg (e_2<s_1)\\ & =(e_1\ge s_2)\wedge (e_2\ge s_1)\end{array}\text{\hspace{1em}By DeMorgan’s Law}$$