Grade 12 : Mathematics - Relations (Set 2) | Concept of Maths

Question 1

1. Let $A = \{1, 2, 3\}$. A relation $R$ is defined on $A$ as $R = \{(1,1), (2,2), (3,3), (1,2), (2,1)\}$. Which of the following is true?

Accepted Answer

C) R is an equivalence relation

Answer

A) R is only reflexive

Answer

B) R is only symmetric

Answer

D) R is reflexive and symmetric but not transitive

Question 2

2. Let $R$ be a relation on the set of real numbers $\mathbb{R}$ defined by aRb if and only if $1 + ab > 0$. Then $R$ is:

Accepted Answer

B) Reflexive and symmetric

Answer

A) Reflexive and transitive

Answer

C) Symmetric and transitive

Answer

D) An equivalence relation

Question 3

3. If $A = \{1, 2, 3, 4\}$, what is the total number of reflexive relations that can be defined on set $A$?

Accepted Answer

A) $2^{12}$

Answer

B) $2^{16}$

Answer

C) $2^4$

Answer

D) $2^8$

Question 4

4. Let $L$ be the set of all lines in a XY-plane and $R$ be a relation in $L$ defined as $R = \{(L_1, L_2) : L_1 \perp L_2\}$. Then $R$ is:

Accepted Answer

D) Symmetric only

Answer

A) Reflexive

Answer

B) Transitive

Answer

C) Equivalence

Question 5

5. Let $R$ be a relation on $\mathbb{Z}$ defined by xRy iff $x - y$ is divisible by 5. The number of distinct equivalence classes is:

Accepted Answer

B) 5

Answer

A) 1

Answer

C) 10

Answer

D) Infinite

Question 6

6. If $R_1$ and $R_2$ are two equivalence relations on a set $A$, then $R_1 \cap R_2$ is:

Accepted Answer

C) An equivalence relation

Answer

A) Only reflexive

Answer

B) Only symmetric

Answer

D) Not necessarily an equivalence relation

Question 7

7. A relation $R$ is defined on $\mathbb{R}$ such that $aRb \iff a \le b^2$. Then $R$ is:

Accepted Answer

D) None of these

Answer

A) Reflexive

Answer

B) Symmetric

Answer

C) Transitive

Question 8

8. The smallest equivalence relation on the set $A = \{1, 2, 3\}$ is:

Accepted Answer

A) {(1,1), (2,2), (3,3)}

Answer

B) {(1,1), (2,2), (3,3), (1,2), (2,1)}

Answer

C) $A 	imes A$

Answer

D) Empty set $\phi$

Question 9

9. If $R = \{(x, y) : x, y \in \mathbb{Z}, x^2 + y^2 \le 4\}$, the domain of $R$ is:

Accepted Answer

B) {-2, -1, 0, 1, 2}

Answer

A) {0, 1, 2}

Answer

C) {-4, -3, ..., 4}

Answer

D) {1, 2}

Question 10

10. Let $R$ be a relation on set $A = \{1, 2, 3\}$ given by $R = \{(1,2), (2,1)\}$. This relation is:

Accepted Answer

C) Symmetric

Answer

A) Reflexive

Answer

B) Transitive

Answer

D) Antisymmetric

Question 11

11. Let $A = \{1, 2, 3\}$. The number of relations containing $(1,2)$ and $(1,3)$ which are reflexive and symmetric but not transitive is:

Accepted Answer

D) 1

Answer

A) 3

Answer

B) 2

Answer

C) 4

Question 12

12. A relation $R$ on set $A$ is called "Circular" if aRb and $bRc \implies cRa$. If a relation is reflexive and circular, it is:

Accepted Answer

A) An equivalence relation

Answer

B) Only symmetric

Answer

C) Only transitive

Answer

D) Not necessarily symmetric

Question 13

13. Let $A$ be the set of all human beings in a town. Relation $R = \{(x, y) : x ext{ is exactly 7cm taller than } y\}$. Then $R$ is:

Accepted Answer

B) Neither reflexive, nor symmetric, nor transitive

Answer

A) Symmetric

Answer

C) Transitive

Answer

D) Reflexive

Question 14

14. If set $A$ has 3 elements, how many symmetric relations can be defined on $A$?

Accepted Answer

C) $2^6$

Answer

A) $2^3$

Answer

B) $2^9$

Answer

D) $2^4$

Question 15

15. For $a, b \in \mathbb{Z}$, define aRb if $ab \ge 0$. Then $R$ is:

Accepted Answer

A) Reflexive and symmetric but not transitive

Answer

B) Equivalence relation

Answer

C) Reflexive and transitive only

Answer

D) Symmetric and transitive only

Question 16

16. Let $R = \{(a, b) : a, b \in \mathbb{R}, a^2 = b^2\}$. This relation is:

Accepted Answer

B) An equivalence relation

Answer

A) Symmetric but not reflexive

Answer

C) Transitive but not symmetric

Answer

D) Anti-symmetric

Question 17

17. The maximum number of equivalence relations on the set $A = \{1, 2, 3\}$ is:

Accepted Answer

D) 5

Answer

A) 2

Answer

B) 3

Answer

C) 4

Question 18

18. In the set $A = \{1, 2, 3, 4, 5\}$, a relation $R$ is defined by $aRb \iff |a^2 - b^2| < 8$. The relation $R$ is:

Accepted Answer

A) Reflexive and symmetric

Answer

B) Symmetric and transitive

Answer

C) Reflexive and transitive

Answer

D) Equivalence relation

Question 19

19. If $R$ is the relation "is a factor of" from $A = \{2, 3, 4\}$ to $B = \{3, 6, 8\}$, then $R$ is:

Accepted Answer

B) {(2,6), (2,8), (3,3), (3,6), (4,8)}

Answer

A) {(2,3), (3,6), (4,8)}

Answer

C) {(2,6), (3,6), (4,8)}

Answer

D) {(3,3), (3,6), (2,8)}

Question 20

20. Let $R$ be a relation on $\mathbb{N}$ defined by $x + 2y = 8$. The range of $R$ is:

Accepted Answer

C) {1, 2, 3}

Answer

A) {2, 4, 6}

Answer

B) {1, 2, 3, 4}

Answer

D) {2, 4}