Problem Solving

goyalsau wrote:HI! Friends I have a problem in Set Theory Formula , Please Help me understand it

(reunion of A, B and C) = (total in A) + (total in B) + (total in C) - (total in exactly 2 groups) - 2(total in all 3 groups)

we have one more formula

# - Intersection ( I m not able to find the sign of intersection )

n ( A U B U C ) = n ( A ) + n ( B ) + n ( C ) - n ( A # B ) - n ( B # C ) - n ( C # A ) + n ( A # B # C )

I know both of the formula are same but i am not to figure out the difference How they are same,
Can any body please help me................

Refer to the above image. If it is not clear then follow me...

Total in A - Blue
Total in B - Red
Total in C - Green

Total in (A ∩ B) = (Red + Blue) = Magenta
Total in (B ∩ C) = (Red + Green) = Yellow
Total in (A ∩ C) = (Blue + Green) = Cyan

Total in (A ∩ B ∩ C) = (Cyan + Magenta + Yellow) = (2*Red + 2*Blue + 2*Green) = White

Only in A and B = Magenta - White = (A ∩ B) - (A ∩ B ∩ C)
Only in B and C = Yellow - White = (B ∩ C) - (A ∩ B ∩ C)
Only in A and C = Cyan - White = (A ∩ C) - (A ∩ B ∩ C)

Therefore only in exactly two groups = Only in A and B + Only in B and C + Only in A and C = (A ∩ B) - (A ∩ B ∩ C) + (B ∩ C) - (A ∩ B ∩ C) + (A ∩ C) - (A ∩ B ∩ C) = (A ∩ B) + (B ∩ C) + (A ∩ C) - 3(A ∩ B ∩ C)

Only in A = Blue - Magenta - Cyan + White = A - (A ∩ B) - (A ∩ C) + (A ∩ B ∩ C)
Only in B = Red - Magenta - Yellow + White = B - (A ∩ B) - (B ∩ C) + (A ∩ B ∩ C)
Only in C = Green - Yellow - Cyan + White = C - (A ∩ C) - (B ∩ C) + (A ∩ B ∩ C)

(White is added as Magenta, Cyan and Yellow also contains white in them, thus white is subtracted twice)

Now, 1st formula:

• (A U B U C) = (total in A) + (total in B) + (total in C) - (total in exactly 2 groups) - 2(total in all 3 groups)
  (A U B U C) = A + B + C - [(A ∩ B) + (B ∩ C) + (A ∩ C) - 3(A ∩ B ∩ C)] - 2*(A ∩ B ∩ C)
  (A U B U C) = A + B + C - (A ∩ B) - (B ∩ C) - (A ∩ C) + (A ∩ B ∩ C)]

Which is same as 2nd formula.

Hope it helps.

@goyalsau: Updated the image. See if it helps.