Set Theory / Venn Diagrams Questions and Answers

Answer: C

If a set has 'n' elements, the total number of its subsets is given by the formula \(2^n\).

We are given that the number of subsets is 64.

So, \(2^n = 64\).

We know that \(2^6 = 64\).

Therefore, n = 6. The set has 6 elements.

Answer: B

The Cartesian product A × B is the set of all ordered pairs (x, y) such that x ∈ A and y ∈ B.

The number of elements in the Cartesian product is given by n(A × B) = n(A) × n(B).

Here, n(A) = 3 and n(B) = 2.

n(A × B) = 3 × 2 = 6.

Answer: B

Let C be the set for cricket and H for hockey.

n(C) = 300, n(H) = 250, n(C ∩ H) = 110.

The number of students who play either game is the union, n(C ∪ H).

n(C ∪ H) = n(C) + n(H) - n(C ∩ H)

n(C ∪ H) = 300 + 250 - 110 = 550 - 110 = 440.

Answer: C

This is an example of the Absorption Law.

The law states that the union of a set A with the intersection of A and another set B results in the original set A.

Visually, A ∩ B is a subset of A. The union of a set and its subset is always the larger set (the superset).

So, A ∪ (A ∩ B) = A.

Answer: A

If sets A and B are disjoint, their intersection is empty, and n(A ∩ B) = 0.

The union of disjoint sets is n(A ∪ B) = n(A) + n(B) = 25 + 20 = 45.

The complement of the union, n((A ∪ B)'), is the number of elements in the universal set but not in the union.

n((A ∪ B)') = n(U) - n(A ∪ B) = 50 - 45 = 5.

Answer: D

Let C be for coffee and T be for tea.

n(C) = 93, n(T) = 47, n(C ∪ T) = 120.

First, find the number of customers who have both: n(C ∩ T).

n(C ∪ T) = n(C) + n(T) - n(C ∩ T)

120 = 93 + 47 - n(C ∩ T)

120 = 140 - n(C ∩ T)

n(C ∩ T) = 140 - 120 = 20.

Number of customers who have only coffee = n(C) - n(C ∩ T) = 93 - 20 = 73.

Answer: B

The notation A ⊂ B means that A is a subset of B, so all elements of A are also in B.

When we take the union of a set with one of its subsets, the result is the larger set (the superset).

Therefore, A ∪ B = B.

Answer: C

Correct answer: 1500

Work in percentages of the whole cohort. The proportion failing at least one subject is

\[49\% + 36\% - 15\% = 70\%\]

subtracting the 15% counted twice. Passing in both subjects means failing neither, so that group is the complement:

\[100\% - 70\% = 30\%\]

This 30% equals 450 students, so

\[\text{Total} = \frac{450}{0.30} = 1500\]

Answer: C

Let F(E) be the percentage of students who failed in English and F(H) be the percentage who failed in Hindi.

Given: n(F(E)) = 49%, n(F(H)) = 36%, n(F(E) ∩ F(H)) = 15%.

The percentage of students who failed in at least one subject is n(F(E) ∪ F(H)).

n(F(E) ∪ F(H)) = n(F(E)) + n(F(H)) - n(F(E) ∩ F(H)) = 49 + 36 - 15 = 70%.

The students who passed in both subjects are those who did not fail in any subject. This is the complement of failing in at least one subject.

Percentage Passed in both = 100% - n(F(E) ∪ F(H)) = 100% - 70% = 30%.

We are given that this number is 675. Let T be the total number of students.

30% of T = 675

0.30 × T = 675

T = 675 / 0.3 = 2250.

The total number of students was 2250.

Answer: C

An empty set, or null set, is a set containing no elements.

A) {x : x² = 9} = {-3, 3}. Not empty.

B) {x : x is an even prime number} = {2}. Not empty (it's a singleton set).

C) {x : x is a month with 32 days}. No month has 32 days. This set is empty.

D) {0}. This is a singleton set containing the element 0. It is not empty.