Question 1

1. Let A = \{1, 2, 3\}. The total number of distinct relations that can be defined on set A is:

A) 9B) $2^3$C) $2^9$D) 81

Answer

✓ Correct Answer: Option C

Explanation: Hint: The number of relations on a set with n elements is $2^{n^2}$. Here n=3.

Question 2

2. A relation R on set A is called an equivalence relation if it is:

A) Reflexive and Symmetric onlyB) Reflexive, Symmetric, and TransitiveC) Symmetric and Transitive onlyD) Reflexive and Transitive only

Question 3

3. Let R be a relation on the set of natural numbers $\mathbb{N}$ defined by xRy if x+2y = 10. The domain of R is:

A) {2, 4, 6, 8, 10}B) {1, 2, 3, 4}C) {2, 4, 6}D) {2, 4, 6, 8}

Answer

✓ Correct Answer: Option D

Explanation: Hint: Find values of $x \in \mathbb{N}$ such that $y = \frac{10-x}{2}$ is also a natural number.

Question 4

4. If A = \{1, 2, 3\}, the number of reflexive relations on A is:

A) $2^6$B) $2^9$C) $2^3$D) 64

Answer

✓ Correct Answer: Option A

Explanation: Hint: The number of reflexive relations is $2^{n^2 - n}$. For n=3, it is $2^{9-3}$.

Question 5

5. Let L be the set of all lines in a plane and R be the relation in L defined by $L_1 R L_2$ if $L_1$ is perpendicular to $L_2$. Then R is:

A) ReflexiveB) TransitiveC) SymmetricD) Equivalence

Answer

✓ Correct Answer: Option C

Explanation: Hint: If $L_1 \perp L_2$, then $L_2 \perp L_1$. However, a line is not perpendicular to itself.

Question 6

6. Consider the relation R in the set \{1, 2, 3, 4\} given by R = \{(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)\}. Then:

A) R is an equivalence relationB) R is reflexive and transitive but not symmetricC) R is reflexive and symmetric but not transitiveD) R is symmetric and transitive but not reflexive

Answer

✓ Correct Answer: Option B

Explanation: Hint: Reflexive pairs (1,1), (2,2), (3,3), (4,4) are all present. (1,2) is present but (2,1) is not.

Question 7

7. Let R be a relation on $\mathbb{R}$ defined by $aRb ⇔ a \le b^2$. Then R is:

A) Neither reflexive, nor symmetric, nor transitiveB) Reflexive but not transitiveC) Symmetric and transitiveD) Reflexive and symmetric

Answer

✓ Correct Answer: Option A

Explanation: Hint: For reflexivity, $0.5 \le (0.5)^2$ is false. For symmetry, $2 \le 3^2$ but $3 \not\le 2^2$ is false (wait, 3 is less than 4, try 2 and 5). For transitivity, $10 \le 4^2$ and $4 \le 3^2$ but $10 \not\le 3^2$.

Question 8

8. The relation R on set A = \{1, 2, 3\} defined by R = \{(1, 1), (2, 2), (3, 3)\} is:

A) Identity relation onlyB) Reflexive relation onlyC) Equivalence relationD) Both A and C

Answer

✓ Correct Answer: Option D

Question 9

9. If R is a relation from set A to set B and S is a relation from set B to set C, then the relation $(S \circ R)^{-1}$ is equal to:

A) $S^{-1} \circ R^{-1}$B) $R^{-1} \circ S^{-1}$C) R \circ SD) S \circ R

Answer

✓ Correct Answer: Option B

Explanation: Hint: The inverse of a composite relation reverses the order of the relations.

Question 10

10. Let R be the relation in the set $\mathbb{Z}$ of integers given by R = \{(a, b) : 2 divides a-b\}. The number of equivalence classes is:

A) 1B) 3C) 2D) Infinite

Answer

✓ Correct Answer: Option C

Question 11

11. Let R be a relation on A = \{1, 2, 3, 4\}. If R = \{(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1), (2, 3), (3, 2)\}, is it transitive?

A) No, because (1,3) is missingB) Yes, it is transitiveC) No, because (4,1) is missingD) No, because (2,2) is present

Answer

✓ Correct Answer: Option A

Explanation: Hint: (1,2) and (2,3) are in R, so (1,3) must be in R for transitivity.

Question 12

12. The number of symmetric relations on a set with n=2 elements is:

A) 4B) 8C) 16D) 2

Answer

✓ Correct Answer: Option B

Question 13

13. Let $R = \{(a, b) : a, b \in \mathbb{R}, a-b + \sqrt{3} is an irrational number\}$. Then R is:

A) ReflexiveB) SymmetricC) TransitiveD) None of these

Answer

✓ Correct Answer: Option D

Explanation: Hint: Test a=sqrt(3), b=0. aRb is true. bRa is 0-sqrt(3)+sqrt(3)=0, which is rational. Not symmetric.

Question 14

14. Let A = \{1, 2, 3\}. The number of equivalence relations containing (1, 2) is:

A) 2B) 1C) 3D) 5

Answer

✓ Correct Answer: Option A

Explanation: Hint: The smallest contains {(1,1),(2,2),(3,3),(1,2),(2,1)}. The largest is A x A.

Question 15

15. A relation R on set A is "Anti-symmetric" if:

A) aRb ⇒ bRaB) aRb and bRa is never possibleC) aRb and bRa ⇒ a = bD) $aRa for all a \in A$

Answer

✓ Correct Answer: Option C

Explanation: Hint: Anti-symmetry means if both directions hold, the elements must be the same.

Question 16

16. Let R be a relation on $\mathbb{N}$ defined by xRy ⇔ x divides y. Then R is:

A) Equivalence RelationB) Partial Order RelationC) Symmetric but not TransitiveD) Not Reflexive

Answer

✓ Correct Answer: Option B

Explanation: Hint: Divisibility is reflexive, anti-symmetric, and transitive.

Question 17

17. If A = \{a, b, c\}, the smallest equivalence relation on A is:

A) $A \times A$B) \{(a, b), (b, c), (c, a)\}C) \phiD) \{(a, a), (b, b), (c, c)\}

Answer

✓ Correct Answer: Option D

Explanation: Hint: The identity relation is the smallest possible equivalence relation.

Question 18

18. Let R = \{(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)\} be a relation on A = \{1, 2, 3, 4\}. The relation is:

A) Not reflexive because (1,1) \notin RB) SymmetricC) TransitiveD) Reflexive

Answer

✓ Correct Answer: Option A

Question 19

19. In the set A = \{1, 2, 3\}, the relation R = \{(1, 2), (2, 1)\} is:

A) Reflexive and SymmetricB) Symmetric onlyC) Transitive onlyD) Equivalence

Answer

✓ Correct Answer: Option B

Explanation: Hint: It is symmetric because (1,2) and (2,1) are present, but lacks (1,1) for reflexivity and transitivity.

Question 20

20. Maximum number of equivalence relations on a set with 3 elements is:

A) 2B) 3C) 5D) 9

Answer

✓ Correct Answer: Option C

Explanation: Hint: The number of equivalence relations on a set of size n is the n-th Bell number. B3 = 5.

Accepted Answer

✓ Correct Answer: Option B

Explanation: Hint: To be an equivalence relation, all three fundamental properties must be satisfied.

Grade 12 : Mathematics - Relations (Set 1)

1. Let A = \{1, 2, 3\}. The total number of distinct relations that can be defined on set A is:

2. A relation R on set A is called an equivalence relation if it is:

3. Let R be a relation on the set of natural numbers $\mathbb{N}$ defined by xRy if x+2y = 10. The domain of R is:

4. If A = \{1, 2, 3\}, the number of reflexive relations on A is:

5. Let L be the set of all lines in a plane and R be the relation in L defined by $L_1 R L_2$ if $L_1$ is perpendicular to $L_2$. Then R is:

6. Consider the relation R in the set \{1, 2, 3, 4\} given by R = \{(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)\}. Then:

7. Let R be a relation on $\mathbb{R}$ defined by $aRb ⇔ a \le b^2$. Then R is:

8. The relation R on set A = \{1, 2, 3\} defined by R = \{(1, 1), (2, 2), (3, 3)\} is:

9. If R is a relation from set A to set B and S is a relation from set B to set C, then the relation $(S \circ R)^{-1}$ is equal to:

10. Let R be the relation in the set $\mathbb{Z}$ of integers given by R = \{(a, b) : 2 divides a-b\}. The number of equivalence classes is:

11. Let R be a relation on A = \{1, 2, 3, 4\}. If R = \{(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1), (2, 3), (3, 2)\}, is it transitive?

12. The number of symmetric relations on a set with n=2 elements is:

13. Let $R = \{(a, b) : a, b \in \mathbb{R}, a-b + \sqrt{3} is an irrational number\}$. Then R is:

14. Let A = \{1, 2, 3\}. The number of equivalence relations containing (1, 2) is:

15. A relation R on set A is "Anti-symmetric" if:

16. Let R be a relation on $\mathbb{N}$ defined by xRy ⇔ x divides y. Then R is:

17. If A = \{a, b, c\}, the smallest equivalence relation on A is:

18. Let R = \{(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)\} be a relation on A = \{1, 2, 3, 4\}. The relation is:

19. In the set A = \{1, 2, 3\}, the relation R = \{(1, 2), (2, 1)\} is:

20. Maximum number of equivalence relations on a set with 3 elements is: