Concept of Venn Diagram to Understand the Relationship Between Sets

easyIf U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and A = {1, 2, 3, 4}, what is n(A′)?

easyIf n(A) = 15, n(B) = 10, and n(A ∩ B) = 5, find n(A ∪ B).

mediumIn a group, 20 people like Tea, 15 like Coffee, and 8 like both. How many like either Tea or Coffee?

mediumIf A ⊂ B, n(A) = 10, and n(B) = 25, what is n(A ∩ B)?

hardGiven n(A)=20, n(B)=25, n(A ∪ B)=40. Find n(A ∩ B′).

hardIf n(U)=50, n(A)=20, n(B)=20, and n(A ∩ B)=5, find n(A′ ∩ B′).

very hardFor three sets A, B, C: n(A)=10, n(B)=10, n(C)=10, n(A∩B)=5, n(B∩C)=5, n(A∩C)=5, n(A∩B∩C)=2. Find n(A ∪ B ∪ C).

Assertion (A): If A = {1, 2} and B = {2, 3}, then n(A ∪ B) = 3.
Reason (R): The formula n(A ∪ B) = n(A) + n(B) − n(A ∩ B) accounts for the overlap element {2}.

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): The complement of the universal set U is the empty set φ.
Reason (R): U′ = U − U = φ, which is consistent with Venn diagram shading.

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 U = {1, 2, ..., 10}, A = {1, 3, 5, 7, 9}, B = {2, 4, 6, 8}. Find: (i) n(A ∩ B) (ii) n(A ∪ B) (iii) n(A′ ∩ B′)

mediumIn a survey of 50 people, 30 like Apples, 25 like Bananas, and 10 like both. Find: (i) Number who like only Apples (ii) Number who like only Bananas (iii) Number who like neither.

hardGiven n(A)=15, n(B)=12, n(A ∪ B)=20. Find: (i) n(A ∩ B) (ii) n(A − B) (iii) n(B − A)

hardA survey of 100 students shows 40 study Physics, 30 study Chemistry, and 25 study Mathematics. 15 study P and C, 10 study C and M, 8 study P and M, and 5 study all three. Find the number of students who study: (i) Exactly one subject (ii) At least two subjects.

very hardIn a class of 60 students, 30 play Football, 25 play Cricket, and 20 play Hockey. 10 play F and C, 8 play C and H, 7 play F and H, and 5 play all three. Calculate: (i) Number of students who play none of the games (ii) Number of students who play exactly two games.

Case Study: A local club has 80 members. 45 members play Badminton, 35 play Table Tennis, and 20 play both. Every member plays at least one of the two games.

If A and B are two sets such that n(A)=20, n(B)=25, and n(A ∪ B)=40, find n(A ∩ B).

In a set of 40 elements, if n(A)=25 and n(B)=20, find the minimum possible value of n(A ∩ B).

If A = {x : x is a prime number < 10} and B = {x : x is an even number < 10}, find n(A ∪ B).

Given n(U)=100, n(A)=50, n(B)=40, n(A ∩ B)=20. Find n(A′ ∩ B′).