Grade 12 : Mathematics - Relations (Set 1) | Concept of Maths

Question 1

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

A) 9

B) \( 2^3 \)

C) \( 2^9 \)

D) 81

Answer

Correct Answer: C

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 only

B) Reflexive, Symmetric, and Transitive

C) Symmetric and Transitive only

D) Reflexive and Transitive only

Answer

Correct Answer: B

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

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: D

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) \( 2^3 \)

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) Reflexive

B) Transitive

C) Symmetric

D) Equivalence

Answer

Correct Answer: C

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 relation

B) \( R \) is reflexive and transitive but not symmetric

C) \( R \) is reflexive and symmetric but not transitive

D) \( R \) is symmetric and transitive but not reflexive

Answer

Correct Answer: B

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 \( aRb \iff a \le b^2 \). Then \( R \) is:

A) Neither reflexive, nor symmetric, nor transitive

B) Reflexive but not transitive

C) Symmetric and transitive

D) Reflexive and symmetric

Answer

Correct Answer: A

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 only

B) Reflexive relation only

C) Equivalence relation

D) Both A and C

Answer

Correct Answer: 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) \( R \circ S \)

D) \( S \circ R \)

Answer

Correct Answer: B

Question 10

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

A) 1

B) 3

C) 2

D) Infinite

Answer

Correct Answer: 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 missing

B) Yes, it is transitive

C) No, because (4,1) is missing

D) No, because (2,2) is present

Answer

Correct Answer: A

Question 12

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

A) 4

B) 8

C) 16

D) 2

Answer

Correct Answer: B

Question 13

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

A) Reflexive

B) Symmetric

C) Transitive

D) None of these

Answer

Correct Answer: D

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) 2

B) 1

C) 3

D) 5

Answer

Correct Answer: A

Question 15

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

A) \( aRb \implies bRa \)

B) \( aRb \text{ and } bRa \text{ is never possible} \)

C) \( aRb \text{ and } bRa \implies a = b \)

D) \( aRa \text{ for all } a \in A \)

Answer

Correct Answer: C

Question 16

16. Let \( R \) be a relation on \( \mathbb{N} \) defined by \( xRy \iff x \text{ divides } y \). Then \( R \) is:

A) Equivalence Relation

B) Partial Order Relation

C) Symmetric but not Transitive

D) Not Reflexive

Answer

Correct Answer: B

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) \( \phi \)

D) \( \{(a, a), (b, b), (c, c)\} \)

Answer

Correct Answer: D

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 \( (1,1) \notin R \)

B) Symmetric

C) Transitive

D) Reflexive

Answer

Correct Answer: A

Question 19

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

A) Reflexive and Symmetric

B) Symmetric only

C) Transitive only

D) Equivalence

Answer

Correct Answer: B

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) 2

B) 3

C) 5

D) 9

Answer

Correct Answer: C

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

Accepted Answer

Correct Answer: A

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

Grade 12 : Mathematics - Relations (Set 1) | Concept of Maths