Problems Using Venn Diagram

easyIf n(A) = 10, n(B) = 12, and n(A ∩ B) = 4, find n(A ∪ B).

easyIn a set U = {1, 2, ..., 20}, if n(A) = 8, find n(A').

mediumIf n(A ∪ B) = 30, n(A) = 20, and n(B) = 15, find n(A ∩ B).

mediumGiven n(A − B) = 12, n(B − A) = 8, and n(A ∩ B) = 5, find n(A ∪ B).

hardIn a group of 50 students, 30 like Tea, 25 like Coffee, and 10 like both. How many like neither?

hardIf n(A) = 20, n(B) = 25, n(C) = 15, n(A∩B)=8, n(B∩C)=7, n(A∩C)=6, n(A∩B∩C)=3, find n(A ∪ B ∪ C).

very hardIf n(U) = 100, n(A) = 40, n(B) = 50, and n(A ∪ B)' = 20, find n(A ∩ B).

Assertion (A): If n(A)=10, n(B)=10, then n(A ∪ B) is always 20.
Reason (R): n(A ∪ B) = n(A) + n(B) − n(A ∩ B), and n(A ∩ B) can be non-zero.

Choose the correct option:
(A) Both A and R are true and R is the correct explanation of A.
(B) Both A and R are true but R is not the correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.

Assertion (A): For any two sets A and B, n(A ∪ B) ≥ n(A).
Reason (R): A is a subset of A ∪ B, therefore the number of elements in A cannot exceed the number of elements in A ∪ B.

Choose the correct option:
(A) Both A and R are true and R is the correct explanation of A.
(B) Both A and R are true but R is not the correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.

mediumGiven n(U) = 60, n(A) = 30, n(B) = 25, n(A ∩ B) = 10. Find: (i) n(A ∪ B) (ii) n(A') (iii) n(A − B)

mediumIn a class, 20 students study Physics, 15 study Chemistry, and 8 study both. Find: (i) Total students studying at least one (ii) Only Physics (iii) Only Chemistry

hardGiven n(A)=25, n(B)=20, n(C)=15, n(A∩B)=10, n(B∩C)=8, n(A∩C)=7, n(A∩B∩C)=5. Find: (i) n(A ∪ B ∪ C) (ii) n(A only) (iii) n(B only)

hardA survey of 100 people shows 60 read Newspaper A, 40 read Newspaper B, and 30 read Newspaper C. 20 read A and B, 15 read B and C, 10 read A and C, and 5 read all three. Find: (i) How many read at least one? (ii) How many read none? (iii) How many read only A?

very hardIn a group of 200 employees, 120 like tea, 100 like coffee, and 80 like juice. 60 like tea and coffee, 50 like coffee and juice, 40 like tea and juice, and 30 like all three. Find: (i) How many like exactly two drinks? (ii) How many like exactly one drink?

Case Study: A sports club has 100 members. 60 play Cricket, 40 play Football, and 30 play Hockey. 20 play Cricket and Football, 15 play Football and Hockey, 10 play Cricket and Hockey, and 5 play all three.

If n(A) = 20, n(B) = 30, n(A ∩ B) = 10, find n(A ∪ B).

In a set of 50 elements, if n(A) = 30 and n(B) = 25, find the minimum value of n(A ∩ B).

If n(A)=15, n(B)=12, n(A ∪ B)=20, find n(A ∩ B).

Given n(A)=20, n(B)=15, n(A ∩ B)=5, find n(A Δ B) where Δ is symmetric difference.