Grade 11 : Mathematics - Sets (Set 3) | Concept of Maths

Question 1

1. Let \( S = \{n \in \mathbb{N} : n \le 100\} \). Let \( A, B, C \) be subsets of \( S \) such that \( A \) is the set of multiples of 2, \( B \) is the set of multiples of 3, and \( C \) is the set of multiples of 5. Find the cardinality of the set \( (A \cap B) \cup (A \cap C) \).

A) 26

B) 20

C) 23

D) 29

Answer

Correct Answer: C

Hint: Use \( |X \cup Y| = |X| + |Y| - |X \cap Y| \). Here \( X = A \cap B \) (multiples of 6) and \( Y = A \cap C \) (multiples of 10). Their intersection is multiples of 30.

Question 2

2. Let \( A = \{ \emptyset, \{ \emptyset \} \} \). Which of the following is the correct Power Set \( P(A) \)?

A) \( \{ \emptyset, \{ \emptyset \} \} \)

B) \( \{ \emptyset, \{ \emptyset \}, \{ \{ \emptyset \} \}, \{ \emptyset, \{ \emptyset \} \} \)

C) \( \{ \emptyset, \{ \{ \emptyset \} \} \)

D) \( \{ \emptyset, \emptyset, \{ \emptyset \} \)

Answer

Correct Answer: B

Hint: A set with \( n \) elements has \( 2^n \) subsets. Here \( n=2 \), so \( P(A) \) must have 4 elements. Elements of \( A \) are \( \emptyset \) and \( \{ \emptyset \} \).

Question 3

3. If \( A \) and \( B \) are two sets such that \( P(A) = P(B) \), then which of the following must be true?

A) \( A = B \)

B) \( A \cap B = \emptyset \)

C) \( A \subseteq B \) but \( A \neq B \)

D) \( n(A) = n(B) \) but \( A \neq B \)

Answer

Correct Answer: A

Hint: The Power Set uniquely determines the set. If every subset of \( A \) is also a subset of \( B \), then the sets themselves must be identical.

Question 4

4. Let \( A = \{ (x, y) : y = e^x, x \in \mathbb{R} \} \) and \( B = \{ (x, y) : y = x, x \in \mathbb{R} \} \). Then \( A \cap B \) is:

A) A singleton set

B) A set containing two elements

C) An infinite set

D) An empty set

Answer

Correct Answer: D

Hint: Consider the function \( f(x) = e^x - x \). Its derivative is \( e^x - 1 \). The minimum value occurs at \( x=0 \), where \( f(0) = 1 \). Since the minimum is positive, \( e^x \) never equals \( x \).

Question 5

5. In a survey of 100 students, the number of students studying various subjects were found to be: Mathematics 60, Physics 45, Chemistry 35. Mathematics and Physics 20, Physics and Chemistry 15, Mathematics and Chemistry 15, all three subjects 10. How many students study exactly two subjects?

A) 30

B) 20

C) 25

D) 50

Answer

Correct Answer: B

Hint: Number of students studying exactly two subjects = \( [n(M \cap P) + n(P \cap C) + n(M \cap C)] - 3n(M \cap P \cap C) \). Calculation: \( (20+15+15) - 3(10) = 50 - 30 = 20 \).

Question 6

6. If \( n(A) = 3 \) and \( n(B) = 6 \), then the minimum and maximum number of elements in \( A \cup B \) are respectively:

A) 3, 9

B) 6, 6

C) 6, 9

D) 0, 9

Answer

Correct Answer: C

Hint: \( n(A \cup B) = n(A) + n(B) - n(A \cap B) \). Max occurs when \( n(A \cap B) = 0 \) (disjoint). Min occurs when \( A \subseteq B \), so \( n(A \cap B) = 3 \).

Question 7

7. Let \( A_n = [0, \frac{1}{n}] \) for each \( n \in \mathbb{N} \). Then the intersection \( \bigcap_{n=1}^{\infty} A_n \) is:

A) \( \{0\} \)

B) \( \emptyset \)

C) \( [0, 1] \)

D) \( (0, 1) \)

Answer

Correct Answer: A

Hint: As \( n \to \infty \), the interval \( [0, 1/n] \) shrinks. The only element present in every such interval is 0.

Question 8

8. The symmetric difference of sets \( A = \{1, 2, 3\} \) and \( B = \{3, 4, 5\} \), denoted by \( A \Delta B \), is:

A) \( \{3\} \)

B) \( \{1, 2, 3, 4, 5\} \)

C) \( \{1, 2, 4, 5\} \)

D) \( \emptyset \)

Answer

Correct Answer: C

Hint: \( A \Delta B = (A - B) \cup (B - A) \). Elements in either A or B but not in both.

Question 9

9. If \( A \) and \( B \) are any two sets, then \( A - (A - B) \) is equivalent to:

A) \( A \cup B \)

B) \( A \cap B \)

C) \( A \)

D) \( B \)

Answer

Correct Answer: B

Hint: \( A - (A \cap B') = A \cap (A \cap B')' = A \cap (A' \cup B) = (A \cap A') \cup (A \cap B) = \emptyset \cup (A \cap B) \).

Question 10

10. Which of the following is always FALSE for any two sets \( A \) and \( B \)?

A) \( P(A) \cap P(B) = P(A \cap B) \)

B) \( P(A) \cup P(B) \subseteq P(A \cup B) \)

C) \( A \Delta B = \emptyset \implies A = B \)

D) \( P(A) \cup P(B) = P(A \cup B) \)

Answer

Correct Answer: D

Hint: The Power Set of the union is generally much larger than the union of the Power Sets. For example, if \( A=\{1\}, B=\{2\} \), \( P(A) \cup P(B) = \{ \emptyset, \{1\}, \{2\} \} \), but \( P(A \cup B) = \{ \emptyset, \{1\}, \{2\}, \{1, 2\} \} \).

Grade 11 : Mathematics - Sets (Set 3) | Concept of Maths