I am wondering about the answer to a particular question which I myself created. Well, I m not wondering about its actual answer, I know it..................What I was wondering is about a mathematical equation which can explain the relation. I even found one Mathematical relation which explains the answer.
This post demonstrates the 'General Equation an Condition of Minimum intersection'.
This post demonstrates the 'General Equation an Condition of Minimum intersection'.
But first let me present the problem.
Lets take a book called 'Space' are 100 chapters. There are 2 students 'A' and 'B'. They read the same book 'Space'. 'A' Randomly reads 75 chapters. 'B' also Randomly reads 75 chapters. The question is, 'minimum' how many chapters did they both studied in common?
--The answer is 50 chapters.
- There are many equations and explainations which solves the problem from different angles and we get the same answer. But my equation was n(A) + n(B) - n(Space) = n(A п B)min.
- Likewise 75 + 75 - 100 = 50.
- I agree that the sets can inter-sect in many ways, but I m interested in knowing and framing the laws and conditions which will have imminent and presise no. of minimum intersection of the sets.
- Here, there is a condition for minimum intersection, the condition is that the sum total of both the sets should have no. of elements which are greater than the n(Space). It is only then that there shall be imminent and presise no. of minimum intersection.
- i.e. If {[ n(A) + n(B) ] > n(Space)} is true, then and only then there shall be imminent and presise no. of minimum intersection.
The above is simple, however the next is somewhat complicated, as it involves the math of 3 sets.
And the problem is:
Lets take the same book called 'Space' which has 100 chapters. There are 3 students 'A' , 'B' and 'C'. They read the same book 'Space'. 'A' Randomly reads 75 chapters. 'B' also Randomly reads 75 chapters. And 'C' also randomly reads say 65 chapters. The question is, 'minimum' how many chapters did all 3 of them studied in common?
--The answer is 15 chapters.
I asked this questions to many people very rarely i got the correct answer. Infact only one IITian Student could give me the right answer, but still i am working to get a fundamental equation which explains the minimum intersection any number of equation.
I saw an equation working on it, The equation is:
- n(A) + n(B) + n(C) - 2 [n(Space)] = n(A п B п C)min.
- Likewise 75 + 75 + 75 - 2(100) = 15.
- Here again there is a condition the condition is similar to the previous one, but not the same, its {[ n(A) + n(B) + n(C) ] > 2(n(Space))}, so the difference is that the sum total of all 3 sets should have no. of elements which are greater than the 2 times the n(Space).
- But remember, if {[ n(A) + n(B) + n(C) ] > 2(n(Space))} then {[ n(A) + n(B) ] > n(Space)}. Which means that n(A п B п C)min can exist only after n(A п B)min exist as n(A п B)min is an integral part of n(A п B п C)min.
Proof: n((A п B) п C)min = n(A п B) + n(C) - n(Space)
= [n(A) + n(B) - n(Space)] - n(C) - n(Space) ..... [as n(A) + n(B) - n(Space) = n(A п B)min]
= n(A) + n(B) + n(C) - 2 [n(Space)] = n(A п B п C)min.
Likewise if we have 'i' no. of sets and if we want to find the minimum intersection of 'i' no. of sets.
- The condition is that [ n(A 1) + n(A 2) + ............... + n(A i) ] > [(i - 1) (n(Space))].
- The general equaton is [ n(A 1) п n(A 2) п ............... п n(A i) ]min = [ n(A 1) + n(A 2) + ............... + n(A i) ] - [(i - 1) (n(Space))] (by Mathematical induction).
Conclusion: If the above conditions are true then there is a certainty that there will be intersection of sets. If not then there will only be a likeliness of intersection.
If you apply the above equation to any number of sets which satisfies the above condition, you get the answer. But I am looking for a simpler explaination and a proper proof of the above.
I even recon that the above relation might be useful for software development techniques specially databases scheme.
note: 'п' stands for intersection.
