Set theory for cat

27) n(A U B U C) = n(A) + n (B)+ n(C) – n(A n B) – n(A n C) – n(B n C) + n(A n B n C)
28) n(A_only) = n(A) – n(A n C) – n(A n B) + n(A U B U C)
To refresh, the union of sets is all elements from all sets. The intersection of sets is only those elements common to all sets. Let’s call our sets A, B, and C. If n = intersection and u = union. The need-to-know formulas:
P(A u B u C) = P(A) + P(B) + P(C) – P(A n B) – P(A n C) – P(B n C) + P(A n B n C)
To find the number of people in exactly one set:
P(A) + P(B) + P(C) – 2P(A n B) – 2P(A n C) – 2P(B n C) + 3P(A n B n C)
To find the number of people in exactly two sets:
P(A n B) + P(A n C) + P(B n C) – 3P(A n B n C)
To find the number of people in exactly three sets:
P(A n B n C)
To find the number of people in two or more sets:
P(A n B) + P(A n C) + P(B n C) – 2P(A n B n C)
To find the number of people in at least one set:

P(A) + P(B) + P(C) – P(A n B) – P(A n C) – P(B n C) + 2 P(A n B n C)
For questions involving set theory, it may be helpful to make a Venn diagram to visualize the solution.
To find the union of all set: (A + B + C + X + Y + Z + O)
Number of people in exactly one set: (A + B + C)
Number of people in exactly two of the sets: (X + Y + Z)
Number of people in exactly three of the sets: O
Number of people in two or more sets: (X + Y + Z + O)