Grade 11 : Mathematics - Relations (Set 10) | Concept of Maths

Question 1

Let P(S) be the power set of S = \{x, y, z\}. A relation R is defined on P(S) such that $A R B ⇔ A \cap B^c = \emptyset$, where $B^c$ is the complement of B. Which of the following describes R?

Accepted Answer

C) Partial Order (Reflexive, Antisymmetric, Transitive)

Answer

A) Symmetric and Reflexive

Answer

B) Equivalence Relation

Answer

D) Only Symmetric

Question 2

Consider the set of all $2 	imes 2$ real matrices $M_2(\mathbb{R})$. Let R be a relation defined by A R B if and only if there exists an invertible matrix P such that $B = P^{-1}AP$. This relation is:

Accepted Answer

B) An Equivalence Relation

Answer

A) Reflexive and Symmetric but not Transitive

Answer

C) Transitive but neither Reflexive nor Symmetric

Answer

D) Antisymmetric and Transitive

Question 3

Let A = \{1, 2, 3, 4, 5\}. What is the total number of distinct symmetric relations that can be defined on A?

Accepted Answer

D) $2^{15}$

Answer

A) $2^{25}$

Answer

B) $2^{10}$

Answer

C) $2^{20}$

Question 4

Let R be a relation on the set of real numbers $\mathbb{R}$ defined by x R y ⇔ 1 + xy > 0. This relation is:

Accepted Answer

A) Reflexive and Symmetric but not Transitive

Answer

B) Reflexive and Transitive but not Symmetric

Answer

C) An Equivalence Relation

Answer

D) Not Reflexive

Question 5

If $R = \{(x, y) : x, y \in \mathbb{Z}, x^2 + y^2 \le 4\}$ and $S = \{(x, y) : x, y \in \mathbb{Z}, x^2 + y^2 \ge 1\}$, find the number of elements in the set $R \cap S$.

Accepted Answer

B) 12

Answer

A) 9

Answer

C) 13

Answer

D) 8

Question 6

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

Accepted Answer

D) Symmetric

Answer

A) Reflexive

Answer

B) Transitive

Answer

C) Equivalence

Question 7

For a set A = \{1, 2, 3\}, let R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)\}. What is the minimum number of ordered pairs that must be added to R to make it an equivalence relation?

Accepted Answer

C) 4

Answer

A) 2

Answer

B) 3

Answer

D) 6

Question 8

Define a relation R on the set of natural numbers $\mathbb{N}$ by x R y if $x^2 - 4xy + 3y^2 = 0$. The relation R is:

Accepted Answer

A) Reflexive but not Symmetric

Answer

B) Symmetric but not Transitive

Answer

C) Reflexive and Transitive

Answer

D) An Equivalence Relation

Question 9

Let R be a relation on $\mathbb{Z}$ defined by a R b if 3a + 4b is divisible by 7. This relation is:

Accepted Answer

B) An Equivalence Relation

Answer

A) Symmetric only

Answer

C) Reflexive but not Transitive

Answer

D) Antisymmetric

Question 10

Let A = \{1, 2, 3\}. Let R = \{(1, 2), (2, 3)\}. The transitive closure of R is:

Accepted Answer

C) \{(1, 2), (2, 3), (1, 3)\}

Answer

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

Answer

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

Answer

D) \{(1, 2), (2, 3), (1, 3), (3, 1)\}