SOLUTION: A, B, and C are sets: n(A)=30; n(B)=32; n(C)=31; n(A ∩ B)=16; n(A ∩ C)=19; n(B ∩ C)=18; and n(A ∩ B ∩ C)=8. Find n(A U B U C) and calculate n(A-(B U C

Algebra ->  Probability-and-statistics -> SOLUTION: A, B, and C are sets: n(A)=30; n(B)=32; n(C)=31; n(A ∩ B)=16; n(A ∩ C)=19; n(B ∩ C)=18; and n(A ∩ B ∩ C)=8. Find n(A U B U C) and calculate n(A-(B U C      Log On


   

Question 1035036: A, B, and C are sets: n(A)=30; n(B)=32; n(C)=31; n(A ∩ B)=16; n(A ∩ C)=19; n(B ∩ C)=18; and n(A ∩ B ∩ C)=8. Find n(A U B U C) and calculate n(A-(B U C)).. Thanks!
Answer by ikleyn(52909) About Me  (Show Source):
You can put this solution on YOUR website!
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Let me introduce more convenient notation.


Let |A| be the number of elements of the set A;
    |B| be the number of elements of the set B;
    |C| be the number of elements of the set C.

Let AB be the intersection of A and B;
    AC be the intersection of A and C;
    BC be the intersection of B and C.

Let |AB| be the number of elements in AB;
    |AC| be the number of elements in AC;
    |BC| be the number of elements in BC.

Let ABC be the intersection of the sets A, B, and C, and
    |ABC| be the number of elements in ABC.

Then there is a remarkable formula:

n(A U B U C) = |A| + |B| + |C| - |AB| - |AC| - |BC| + |ABC|.

For its proof see the lesson Advanced problems on counting elements in sub-sets of a given finite set in this site.

Now apply the formula. You have everything to calculate the right side. It is 

30 + 32 + 31 - 16 - 19 - 18 + 8 = 48.

Answer. n(A U B U C) = 48.