Using only the functions $\max$ and $\min$ and arithmetic operations (no if clauses), express the amount of possible overlap between two intervals $[a_1,a_2]$ and $[b_1,b_2]$, where $a_1,a_2,b_1,b_2$ are arbitrary real numbers with $a_1<a_2$ and $b_1<b_2$.
  • The $\min$ and $\max$ functions are applied to two or more numbers, and return the smallest and largest of them respectively, e.g. $\max(-1,-3)=-1.$

Think about intervals on the real number line. Sketch $[a_1,a_2]$ and $[b_1,b_2]$ when these overlap?

What is the left boundary of the overlap in terms of min/max of the 4 numbers?

What about the right boundary?

What if there is no overlap? Remember you can only use min/max functions.

Assume that intervals overlap, partially or fully, like in the figure above. We can notice that the smallest value in this intersection is always $\max(a_1, b_1),$ and the largest value in this intersection is $\min(a_2,b_2).$ Hence, the amount of overlap of these intervals can be expressed as $\min(a_2,b_2)-\max(a_1, b_1).$

There will be some overlap only if this value is positive, hence the final formula is $\max(0,\min(a_2,b_2)-\max(a_1, b_1)).$

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