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# Practice Paper 1 Question 5

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. $$\max(-1,-3)=-1.$$

## Hints

• Hint 1
Think about intervals on the real number line. Sketch $$[a_1,a_2]$$ and $$[b_1,b_2]$$ when these overlap?
• Hint 2
What is the left boundary of the overlap in terms of min/max of the 4 numbers?
• Hint 3
What about the right boundary?
• Hint 4
What if there is no overlap? Remember you can only use min/max functions.

## Solution

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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