1. If n(A)=15, n(B)=10, and n(A ∩ B)=4, find n(A ∪ B).
Answer: 21. Formula: 15 + 10 - 4 = 21.
2. In a group of 40 students, 25 play football and 20 play basketball. If 10 play both, how many play NEITHER?
Answer: 5. Union = 25 + 20 - 10 = 35. Neither = 40 (Total) - 35 = 5.
3. If n(A)=30 and n(A ∩ B)=12, find n(A - B) (A only).
Answer: 18. Start with the whole set A (30) and subtract the overlap (12).
4. A survey of 100 people found 60 like Apple and 50 like Samsung. If everyone likes at least one, how many like BOTH?
Answer: 10. 60 + 50 = 110. Since there are only 100 people, 10 must have been counted twice.
5. What does the region outside both circles in a Venn diagram represent?
Answer: (A ∪ B)' or those who satisfy Neither condition.
6. If n(A)=12, n(B)=15, and they are disjoint, find n(A ∪ B).
Answer: 27. Disjoint means intersection is 0, so just add them: 12 + 15 = 27.
7. In a class of 30, 18 like Coffee. How many don't like Coffee?
Answer: 12. n(C') = n(U) - n(C) = 30 - 18 = 12.
8. If n(A ∪ B)=50, n(A)=30, and n(B)=30, find n(A ∩ B).
Answer: 10. 50 = 30 + 30 - x → 50 = 60 - x → x = 10.
9. "Only B" is represented by which set operation?
Answer: B - A.
10. If n(A)=20 and A ⊆ B, and n(B)=30, find n(A ∪ B).
Answer: 30. If A is inside B, their union is simply B.
11. A shop has 50 shirts. 30 are Red, 25 have buttons. If 15 are Red with buttons, how many are neither?
Answer: 10. Union = 30 + 25 - 15 = 40. Neither = 50 - 40 = 10.
12. True or False: n(A) + n(B) is always equal to n(A ∪ B).
Answer: False. Only true if the sets are disjoint (no overlap).
13. In a Venn diagram, where do you write the number of elements shared by two sets?
Answer: In the overlapping (center) region.
14. If n(U)=100 and n(A ∪ B)=85, what is n(A ∪ B)'?
Answer: 15. (100 - 85).
15. A group of 15 friends: 10 like Pizza, 8 like Burgers. How many like BOTH? (Assume everyone likes at least one).
Answer: 3. 10 + 8 - 15 = 3.
16. If n(A)=20 and n(A ∩ B)=20, what can you say about A?
Answer: A ⊆ B. All of A is inside the overlap.
17. n(A) = 40. n(A only) = 25. Find n(A ∩ B).
Answer: 15. (40 - 25).
18. In a survey, "Total" usually refers to which set?
Answer: The Universal Set (U).
19. If n(A ∩ B) increases while n(A) and n(B) stay the same, does n(A ∪ B) increase or decrease?
Answer: Decrease. Because you are subtracting a larger number in the formula.
20. What is the first value you should try to fill in a Venn diagram?
Answer: The Intersection (Both).