Grade 11 : Mathematics - Relations (Set 8) | Concept of Maths

Question 1

Two robotic arms are programmed using coordinate pairs. If the set of x-coordinates is $A = \{x \in \mathbb{Z} : 1 < x \le 3\}$ and the set of y-coordinates is B = \{5, 6\}, determine the Cartesian product $A 	imes B$.

Accepted Answer

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

Answer

B) \{(1, 5), (1, 6), (2, 5), (2, 6), (3, 5), (3, 6)\}

Answer

C) \{(2, 5), (3, 6)\}

Answer

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

Question 2

A central database server manages connections between two clusters of nodes, P and Q. If n(P) = 3 and n(Q) = 4, how many distinct non-empty relations can be defined from Cluster P to Cluster Q?

Accepted Answer

B) $2^{12} - 1$

Answer

A) $2^{7} - 1$

Answer

C) $12^2 - 1$

Answer

D) $2^{12}$

Question 3

Consider a signal processing relation R defined on the set of natural numbers $\mathbb{N}$ by R = \{(x, y) : x + 2y = 10\}. What is the range of this relation?

Accepted Answer

C) \{1, 2, 3, 4, 5\} is not possible; specifically \{1, 2, 3, 4\}

Answer

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

Answer

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

Answer

D) \{1, 2, 3, 4\} because y must be natural.

Question 4

In a geometric simulation, a relation R is defined on the set S = \{0, 1, 2, 3, 4, 5\} by $R = \{(x, y) : x^2 + y^2 = 25\}$. Find the domain of R.

Accepted Answer

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

Answer

B) \{3, 4, 5\}

Answer

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

Answer

D) \{0, 5\}

Question 5

Let A = \{1, 2, 3, 4\}. A relation R on A is given by R = \{(1, 2), (2, 2), (3, 1), (4, 3)\}. Identify the inverse relation $R^{-1}$.

Accepted Answer

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

Answer

A) \{(2, 1), (2, 2), (1, 3), (4, 3)\}

Answer

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

Answer

C) \{(2, 1), (1, 2), (1, 3), (3, 4)\}

Question 6

A relation R is defined on the set of integers $\mathbb{Z}$ by aRb if and only if a - b is divisible by 5. Which of the following properties does this relation satisfy?

Accepted Answer

B) Reflexive, Symmetric, and Transitive (Equivalence)

Answer

A) Reflexive only

Answer

C) Symmetric and Transitive but not Reflexive

Answer

D) Reflexive and Symmetric but not Transitive

Question 7

Let L be the set of all lines in a 2D plane. A relation R is defined on L such that $l_1 R l_2$ if $l_1$ is perpendicular to $l_2$. This relation is:

Accepted Answer

C) Symmetric

Answer

A) Reflexive

Answer

B) Transitive

Answer

D) Equivalence

Question 8

If a set S contains 3 elements, how many reflexive relations can be formed on S?

Accepted Answer

B) $2^6$

Answer

A) $2^9$

Answer

C) $2^3$

Answer

D) 64

Question 9

Two ordered pairs (x+y, 1) and (3, x-y) are equal. What is the relation between x and y?

Accepted Answer

B) x=2, y=1

Answer

A) x=1, y=2

Answer

C) x=3, y=0

Answer

D) x=2, y=2

Question 10

A mapping R for a data set is defined by $R = \{(x, y) : y = |x-1|, x \in \{-2, -1, 0, 1, 2\}\}$. Determine the range of the mapping.

Accepted Answer

C) \{0, 1, 2, 3\}

Answer

A) \{1, 2, 3\}

Answer

B) \{0, 1, 2\}

Answer

D) \{-3, -2, -1, 0, 1\}