Grade 11 Mathematics Quiz: Relations (Set 6)

Question 1

Let set A represent the set of prime factors of 210 and set B represent the set of divisors of 10. Find the total number of elements in the Cartesian product A \ imes B.

Accepted Answer

C) 16

Answer

A) 12

Answer

B) 15

Answer

D) 20

Question 2

In a specialized logistics network, a relation R is defined on the set X = \{2, 3, 4, 5, 6\} such that $(a, b) \in R$ if and only if a and b are coprime (i.e., gcd(a, b) = 1). Which of the following ordered pairs belongs to R?

Accepted Answer

B) (3, 5)

Answer

A) (4, 6)

Answer

C) (2, 4)

Answer

D) (3, 6)

Question 3

If n(A) = p and n(B) = q, the total number of non-empty relations that can be defined from A to B is given by:

Accepted Answer

A) $2^{pq} - 1$

Answer

B) $2^{p+q}$

Answer

C) $p^q - 1$

Answer

D) $2^{pq}$

Question 4

A relation R is defined on the set of natural numbers $\mathbb{N}$ as $R = \{(x, y) : x + 2y = 10, x, y \in \mathbb{N}\}$. Determine the Range of R.

Accepted Answer

D) \{1, 2, 3, 4\}

Answer

A) \{2, 4, 6, 8\}

Answer

B) \{1, 2, 3, 4, 5\}

Answer

C) \{8, 6, 4, 2\}

Question 5

Let R be a relation on the set $\mathbb{Z}$ defined by $R = \{(a, b) : a^2 = b^2\}$. This relation is:

Accepted Answer

B) Reflexive, Symmetric, and Transitive

Answer

A) Only Reflexive

Answer

C) Only Symmetric

Answer

D) Anti-symmetric

Question 6

Suppose A = \{x, y, z\} and B = \{1, 2\}. Find the number of relations from A to B that contain exactly 5 elements.

Accepted Answer

C) 6

Answer

A) 3

Answer

B) 12

Answer

D) 64

Question 7

Let R be a relation defined by $R = \{(x, y) : y = x + \rac{6}{x}, x \in \{1, 2, 3, 6\}, y \in \mathbb{N}\}$. The roster form of R is:

Accepted Answer

A) \{(1, 7), (2, 5), (3, 5), (6, 7)\}

Answer

B) \{(1, 6), (2, 3), (3, 2), (6, 1)\}

Answer

C) \{(1, 7), (2, 5)\}

Answer

D) \{(7, 1), (5, 2), (5, 3), (7, 6)\}

Question 8

Find the Domain of the relation $R = \{(x, y) : x^2 + y^2 = 25, x, y \in \mathbb{W}\}$ (where $\mathbb{W}$ is the set of whole numbers).

Accepted Answer

D) \{0, 3, 4, 5\}

Answer

A) \{0, 1, 2, 3, 4, 5\}

Answer

B) \{5, 4, 3, 0\}

Answer

C) \{3, 4, 5\}

Question 9

In a data science experiment, set S has 4 sensors and set M has k monitors. If there are 1024 possible relations from S to M, find the value of k.

Accepted Answer

C) 2.5 (Invalid) -> Correction: Let k be 2.5? No. Let's re-evaluate: $2^{4k} = 1024$.

Answer

A) 2

Answer

B) 4

Answer

D) 10

Question 10

If R is a relation on A = \{1, 2, 3\} defined as R = \{(1,1), (2,2), (3,3), (1,2), (2,1)\}, which property does it satisfy?

Accepted Answer

B) Symmetric and Reflexive

Answer

A) Anti-symmetric

Answer

C) Transitive but not Reflexive

Answer

D) None of the above