Finding regions of overlap

Am I missing something?  It seems quite easy to me:

Suppose the intervals are (l1, u1), (l2, u2), ..., (ln, un).  The
intersection, if non-empty, would be (max(l1, ..., ln), min(u1, ..., un)),
and the union (min(l1, ..., ln), max(u1, ..., un)).