# What is the Shortest possible formula to find the intersection between a set of two ranges of number

How to find the intersection between a set of two ranges of number.

let me explain the question with an example,

{2,3} {3,8} would result to 0
{2,5} {3,8} would result to 2
{3,6} {3,8} would result to 3
{4,5} {3,8} would result to 1
{4,5} {3,8} would result to 1
{8,9} {3,8} would result to 0


Note:This could be easily done with series of if in a computer application, I am hungry for a mathematical solution!

• By "intersection", you mean the number of "units" of overlap between the two given intervals? Dec 29, 2010 at 9:08
• What are "tow ranges"? Dec 29, 2010 at 9:13
• @ J.M yes , and @ Pete L.Clark I mean set of two numbers Dec 29, 2010 at 9:15
• There is a great grief about this notation of yours, $(3,8)$ usually indicates all the real numbers which are strictly greater than $3$ and strictly smaller than $8$. Not just the integers, which is what seems to be your meaning. Dec 29, 2010 at 9:20
• @Asaf Karagila I have changed to {,} I am new so not sure this is correct ,thanks Dec 29, 2010 at 9:41

If I understood you correctly, you have $\{x,x+1,\ldots,x+n\}$ and $\{y,y+1,\ldots,y+k\}$ and granted $x,y$ are integers you want to find out the number of elements in the intersections (i.e. how many numbers appear in both sets).
Well, take: $$\max\{0,\min\{y+k,x+n\}-\max\{x,y\}\}$$ and that should be it.