When do two intervals overlap?

For a scheduling tool I encountered the following problem. For a set of intervals generate a list of a pairs that overlap. For those pairs (A,B) I can generate a constraint (Note: we are smart and do not generate both x_A+x_B≤1 and x_B+x_A≤1) The test if two intervals A=[a₁,a₂], B=[b₁,b₂] overlap is:

or

depending on borderline cases (the last version is called the “strict” version in the following). Here is the demonstration:

Note that if the intervals are determined endogenously in the model, we often write “no-overlap” constraints for two intervals A=[a₁,a₂], B=[b₁,b₂] as:

I.e. B starts after A ends or A starts after B ends. Indeed, we can see that

which derives our second “strict” form of the overlap conditions.