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) 26B) 20C) 23D) 29

Answer

✓ Correct Answer: Option C

Explanation: 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 \} \}

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 = BB) $A \cap B = \emptyset$C) $A \subseteq B$ but $A \neq B$D) n(A) = n(B) but $A \neq B$

Answer

✓ Correct Answer: Option A

Explanation: 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 setB) A set containing two elementsC) An infinite setD) An empty set

Answer

✓ Correct Answer: Option D

Explanation: 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) 30B) 20C) 25D) 50

Answer

✓ Correct Answer: Option B

Explanation: 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, 9B) 6, 6C) 6, 9D) 0, 9

Answer

✓ Correct Answer: Option C

Explanation: 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) \emptysetC) [0, 1]D) (0, 1)

Answer

✓ Correct Answer: Option A

Explanation: 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: Option C

Explanation: 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) AD) B

Answer

✓ Correct Answer: Option B

Explanation: 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 ⇒ A = BD) $P(A) \cup P(B) = P(A \cup B)$

Answer

✓ Correct Answer: Option D

Explanation: 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 $P(A \cup B) = \{ \emptyset, \{1\}, \{2\}, \{1, 2\} \}$.

Accepted Answer

✓ Correct Answer: Option B

Explanation: 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 \}.

Grade 11 : Mathematics - Sets (Set 3)

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 $(A \cap B) \cup (A \cap C)$.

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

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

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:

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?

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

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

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

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

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