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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!

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By "intersection", you mean the number of "units" of overlap between the two given intervals? –  J. M. Dec 29 '10 at 9:08
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What are "tow ranges"? –  Pete L. Clark Dec 29 '10 at 9:13
    
@ J.M yes , and @ Pete L.Clark I mean set of two numbers –  Thunder Dec 29 '10 at 9:15
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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. –  Asaf Karagila Dec 29 '10 at 9:20
    
@Asaf Karagila I have changed to {,} I am new so not sure this is correct ,thanks –  Thunder Dec 29 '10 at 9:41
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1 Answer

up vote 8 down vote accepted

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.

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well the result is not a set but a number , that represents the value in the intersection set. –  Thunder Dec 29 '10 at 9:40
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@Thunder: Yes, and Asaf's construction does return the number you (seem to) want. –  J. M. Dec 29 '10 at 9:50
    
yes this the correct answer ... thanks –  Thunder Dec 29 '10 at 9:52
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