Follow this formula for two sets:
Let’s call our sets A, B
All (A U B) = A + B – Both (A ∩ B) + None
Formulas for Three Sets:
Let’s call our sets A, B, and C.
If n = intersection and u = union.
Here are 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.
What is "Venn Diagram":
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)
THREE WAY VENN DIAGRAMS: