Question 1

1. 1. If the ordered pairs (x+2, 4) and (5, 2y) are equal, find the values of x and y.

A) x=3, y=4B) x=7, y=2C) x=3, y=2D) x=5, y=4

Answer

✓ Correct Answer: Option C

Question 2

2. 2. Let A = \{1, 2\} and B = \{3, 4\}. How many relations are possible from set A to set B?

A) 4B) 16C) 8D) 32

Question 3

3. 3. If n(A) = 3 and n(B) = 4, then the number of elements in $(A \times B) \cap (B \times A)$ is at most:

A) 9B) 12C) 16D) 7

Answer

✓ Correct Answer: Option A

Explanation: Hint: The intersection $(A \times B) \cap (B \times A)$ is equal to $(A \cap B) \times (B \cap A)$. The maximum number of common elements in sets of size 3 and 4 is 3. So, $3 \times 3 = 9$.

Question 4

4. 4. Find the range of the relation $R = \{ (x, x^2) : x is a prime number less than 10 \}$.

A) \{1, 4, 9, 25, 49\}B) \{4, 9, 25, 49, 81\}C) \{2, 3, 5, 7\}D) \{4, 9, 25, 49\}

Answer

✓ Correct Answer: Option D

Explanation: Hint: Prime numbers less than 10 are 2, 3, 5, 7. Their squares are 4, 9, 25, 49.

Question 5

5. 5. If $A \times B = \{ (a, 1), (b, 3), (a, 3), (b, 1), (a, 2), (b, 2) \}$, find set A.

A) \{1, 2, 3\}B) \{a, b\}C) \{a, b, 1, 2, 3\}D) \{a, 1, b, 3\}

Answer

✓ Correct Answer: Option B

Explanation: Hint: Set A consists of all the first elements of the ordered pairs in $A \times B$.

Question 6

6. 6. Let R be a relation on $\mathbb{N}$ defined by R = \{ (x, y) : x + 2y = 10 \}. The domain of R is:

A) \{2, 4, 6, 8, 10\}B) \{1, 2, 3, 4\}C) \{2, 4, 6, 8\}D) \{1, 3, 5, 7, 9\}

Answer

✓ Correct Answer: Option C

Explanation: Hint: Since $y \in \mathbb{N}$, $y \ge 1$. If y=1, x=8; y=2, x=6; y=3, x=4; y=4, x=2. If y=5, x=0 (not in $\mathbb{N}$).

Question 7

7. 7. If $R = \{ (x, y) : x, y \in \mathbb{Z}, x^2 + y^2 = 9 \}$, which of the following is NOT an element of R?

A) $(1, \sqrt{8})$B) (0, 3)C) (-3, 0)D) (0, -3)

Answer

✓ Correct Answer: Option A

Explanation: Hint: Elements must be integers ($\mathbb{Z}$). $\sqrt{8}$ is not an integer.

Question 8

8. 8. Let A = \{1, 2, 3, 4, 5, 6\}. Define a relation R from A to A by R = \{ (x, y) : y = x + 1 \}. The range of R is:

A) \{1, 2, 3, 4, 5, 6\}B) \{1, 2, 3, 4, 5\}C) \{2, 3, 4, 5\}D) \{2, 3, 4, 5, 6\}

Answer

✓ Correct Answer: Option D

Explanation: Hint: For $x \in \{1, 2, 3, 4, 5\}$, y values are \{2, 3, 4, 5, 6\}. If x=6, y=7 \notin A.

Question 9

9. 9. If $A \times B$ has 6 elements and three of them are (1, 3), (2, 5), (3, 3), then find $A \times B$.

A) \{(1,3), (1,5), (2,3), (2,5), (3,5)\}B) \{(1,3), (1,5), (2,3), (2,5), (3,3), (3,5)\}C) \{(1,3), (2,3), (3,3), (5,1), (5,2), (5,3)\}D) \{(3,1), (3,2), (3,3), (5,1), (5,2), (5,3)\}

Answer

✓ Correct Answer: Option B

Explanation: Hint: First elements give A = \{1, 2, 3\}, second elements give B = \{3, 5\}. $n(A \times B) = 3 \times 2 = 6$.

Question 10

10. 10. The Cartesian product $A \times A$ has 9 elements, among which are found (-1, 0) and (0, 1). Find the set A.

A) \{-1, 1\}B) \{0, 1\}C) \{-1, 0, 1\}D) \{-1, 0, 1, 2\}

Answer

✓ Correct Answer: Option C

Explanation: Hint: Since $n(A \times A) = 9$, n(A) = 3. Elements gathered from the pairs are -1, 0, 1.

Question 11

11. 11. Let R be a relation from A = \{2, 3, 4, 5\} to B = \{3, 6, 7, 10\} defined by x R y ⇔ x divides y. Find R in Roaster form.

A) \{(2,6), (2,10), (3,3), (3,6), (5,10)\}B) \{(2,6), (2,10), (3,6), (5,10)\}C) \{(3,3), (6,2), (10,2), (10,5)\}D) \{(2,3), (2,6), (2,7), (2,10)\}

Answer

✓ Correct Answer: Option A

Explanation: Hint: Check which x from A divides y from B. 2 divides 6, 10; 3 divides 3, 6; 5 divides 10.

Question 12

12. 12. If $R^{-1} = \{ (2, 1), (4, 3), (6, 5) \}$, then the domain of the original relation R is:

A) \{2, 4, 6\}B) \{1, 3, 5\}C) \{1, 2, 3, 4, 5, 6\}D) \{2, 1, 4, 3, 6, 5\}

Answer

✓ Correct Answer: Option B

Question 13

13. 13. Let A = \{1, 2\}. The number of non-empty relations on A is:

A) 16B) 4C) 15D) 12

Answer

✓ Correct Answer: Option C

Explanation: Hint: Total relations = $2^{n(A) \times n(A)} = 2^4 = 16$. Non-empty = 16 - 1 = 15.

Question 14

14. 14. If (x+y, x-y) = (4, 2), then the values of x and y are:

A) x=4, y=2B) x=2, y=1C) x=3, y=0D) x=3, y=1

Answer

✓ Correct Answer: Option D

Explanation: Hint: Solve the linear system: x+y=4 and x-y=2. Adding them gives 2x=6 ⇒ x=3.

Question 15

15. 15. Find the domain of $R = \{ (x, y) : y = |x-1|, x \in \mathbb{Z}, |x| \le 2 \}$.

A) \{-2, -1, 0, 1, 2\}B) \{0, 1, 2, 3\}C) \{-2, 2\}D) \{1, 2, 3\}

Answer

✓ Correct Answer: Option A

Explanation: Hint: $|x| \le 2$ for integers means $x \in \{-2, -1, 0, 1, 2\}$.

Question 16

16. 16. If n(A) = p and n(B) = q, then the total number of non-void relations from A to B is:

A) pqB) $2^{pq} - 1$C) $2^{pq}$D) $p^q - 1$

Answer

✓ Correct Answer: Option B

Explanation: Hint: "Non-void" means non-empty. Total relations is $2^{n(A)n(B)}$.

Question 17

17. 17. Let A = \{1, 2, 3\}. Which of the following relations on A is the Identity Relation?

A) \{(1,1), (2,2)\}B) \{(1,1), (2,2), (3,3), (1,2)\}C) \{(1,1), (2,2), (3,3)\}D) \{(3,3), (2,2), (1,3)\}

Answer

✓ Correct Answer: Option C

Explanation: Hint: Identity relation $I_A$ contains only pairs (a, a) for every $a \in A$.

Question 18

18. 18. If A = \{x, y, z\} and B = \{1, 2\}, then $n(A \times B)$ is:

A) 6B) 5C) 9D) 8

Answer

✓ Correct Answer: Option A

Question 19

19. 19. The relation $R = \{ (x, y) : x, y \in \mathbb{N}, 2x + y = 41 \}$. The range of R is:

A) \{1, 2, 3, \dots, 20\}B) \{1, 3, 5, \dots, 39\}C) \{2, 4, 6, \dots, 40\}D) \{1, 3, 5, \dots, 39\}

Answer

✓ Correct Answer: Option D

Explanation: Hint: y = 41 - 2x. Since $x \ge 1$, y is odd and less than 41. Range is all odd numbers from 1 to 39.

Question 20

20. 20. If $A \subset B$, then $A \times C$ is a subset of:

A) $C \times B$B) $B \times C$C) $A \times B$D) $C \times A$

Answer

✓ Correct Answer: Option B

Explanation: Hint: If every element of A is in B, then every pair in $A \times C$ must be in $B \times C$.

Grade 11 : Mathematics - Relations (Set 3)

1. <b>1. If the ordered pairs (x+2, 4) and (5, 2y) are equal, find the values of x and y.</b>

2. <b>2. Let A = \{1, 2\} and B = \{3, 4\}. How many relations are possible from set A to set B?</b>

3. <b>3. If n(A) = 3 and n(B) = 4, then the number of elements in $(A \times B) \cap (B \times A)$ is at most:</b>

4. <b>4. Find the range of the relation $R = \{ (x, x^2) : x is a prime number less than 10 \}$.</b>

5. <b>5. If $A \times B = \{ (a, 1), (b, 3), (a, 3), (b, 1), (a, 2), (b, 2) \}$, find set A.</b>

6. <b>6. Let R be a relation on $\mathbb{N}$ defined by R = \{ (x, y) : x + 2y = 10 \}. The domain of R is:</b>

7. <b>7. If $R = \{ (x, y) : x, y \in \mathbb{Z}, x^2 + y^2 = 9 \}$, which of the following is NOT an element of R?</b>

8. <b>8. Let A = \{1, 2, 3, 4, 5, 6\}. Define a relation R from A to A by R = \{ (x, y) : y = x + 1 \}. The range of R is:</b>

9. <b>9. If $A \times B$ has 6 elements and three of them are (1, 3), (2, 5), (3, 3), then find $A \times B$.</b>

10. <b>10. The Cartesian product $A \times A$ has 9 elements, among which are found (-1, 0) and (0, 1). Find the set A.</b>

11. <b>11. Let R be a relation from A = \{2, 3, 4, 5\} to B = \{3, 6, 7, 10\} defined by x R y ⇔ x divides y. Find R in Roaster form.</b>

12. <b>12. If $R^{-1} = \{ (2, 1), (4, 3), (6, 5) \}$, then the domain of the original relation R is:</b>

13. <b>13. Let A = \{1, 2\}. The number of non-empty relations on A is:</b>

14. <b>14. If (x+y, x-y) = (4, 2), then the values of x and y are:</b>

15. <b>15. Find the domain of $R = \{ (x, y) : y = |x-1|, x \in \mathbb{Z}, |x| \le 2 \}$.</b>

16. <b>16. If n(A) = p and n(B) = q, then the total number of non-void relations from A to B is:</b>

17. <b>17. Let A = \{1, 2, 3\}. Which of the following relations on A is the Identity Relation?</b>

18. <b>18. If A = \{x, y, z\} and B = \{1, 2\}, then $n(A \times B)$ is:</b>

19. <b>19. The relation $R = \{ (x, y) : x, y \in \mathbb{N}, 2x + y = 41 \}$. The range of R is:</b>

20. <b>20. If $A \subset B$, then $A \times C$ is a subset of:</b>