Grade 11 : Mathematics - Relations (Set 1) | Concept of Maths

Question 1

1. If set $A = \{1, 2\}$ and set $B = \{a, b\}$, which of the following represents the Cartesian product $A \times B$?

A) $\{(1, a), (1, b), (2, a), (2, b)\}$

B) $\{(a, 1), (a, 2), (b, 1), (b, 2)\}$

C) $\{(1, 1), (2, 2), (a, a), (b, b)\}$

D) $\{(1, a), (2, b)\}$

Answer

Correct Answer: A

The Cartesian product $A \times B$ is the set of all ordered pairs $(x, y)$ such that $x \in A$ and $y \in B$. For $A = \{1, 2\}$ and $B = \{a, b\}$: $A \times B = \{(1, a), (1, b), (2, a), (2, b)\}$.

Question 2

2. If $n(A) = 3$ and $n(B) = 2$, what is the total number of possible relations from set $A$ to set $B$?

A) $6$

B) $32$

C) $64$

D) $12$

Answer

Correct Answer: C

The number of elements in $A \times B$ is $n(A) \times n(B) = 3 \times 2 = 6$. The total number of relations from $A$ to $B$ is the number of subsets of $A \times B$, which is $2^{n(A \times B)}$. Total relations = $2^6 = 64$.

Question 3

3. Find the domain of the relation $R = \{(10, 1), (20, 2), (30, 3)\}$.

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

B) $\{10, 20, 30\}$

C) $\{10, 1, 20, 2, 30, 3\}$

D) $\{(10, 1), (20, 2), (30, 3)\}$

Answer

Correct Answer: B

The domain of a relation is the set of all first components of the ordered pairs. Domain = $\{10, 20, 30\}$.

Question 4

4. A relation $R$ is defined from $A = \{1, 2, 3\}$ to $B = \{4, 5\}$ by $R = \{(1, 4), (2, 5), (3, 4)\}$. What is the range of $R$?

Question 5

5. If $R = \{(x, y) : y = x + 2\}$ is a relation on the set $A = \{1, 2, 3, 4\}$, which of the following is an element of $R$?

A) $(1, 3)$

B) $(3, 1)$

C) $(4, 6)$

D) $(2, 2)$

Answer

Correct Answer: A

We test each option using the rule $y = x + 2$ where $x, y \in A$: 1. $(1, 3): 3 = 1 + 2$ (True, and both $1, 3 \in A$). 2. $(3, 1): 1 = 3 + 2$ (False). 3. $(4, 6): 6 = 4 + 2$ (True, but $6 \notin A$). 4. $(2, 2): 2 = 2 + 2$ (False). Thus, $(1, 3) \in R$.

Question 6

6. If $R$ is a relation from $A$ to $B$, then $R$ is mathematically defined as:

A) $R = A \cup B$

B) $R \subseteq A \cup B$

C) $R \subseteq A \times B$

D) $R = A \times B$

Answer

Correct Answer: C

By definition, a relation $R$ from set $A$ to set $B$ is any subset of the Cartesian product $A \times B$. Hence, $R \subseteq A \times B$.

Question 7

7. What is the inverse relation $R^{-1}$ if $R = \{(p, 1), (q, 2), (r, 3)\}$?

A) $\{(1, p), (2, r), (3, q)\}$

B) $\{(1, p), (2, q), (3, r)\}$

C) $\{(p, 1), (q, 2), (r, 3)\}$

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

Answer

Correct Answer: B

The inverse relation $R^{-1}$ is obtained by swapping the components of each ordered pair in $R$. $R^{-1} = \{(1, p), (2, q), (3, r)\}$.

Question 8

8. Let $A = \{5, 6\}$. Which of the following is the identity relation $I_A$ on set $A$?

A) $\{(5, 6), (6, 5)\}$

B) $\{(5, 5), (5, 6), (6, 6)\}$

C) $\{(6, 6)\}$

D) $\{(5, 5), (6, 6)\}$

Answer

Correct Answer: D

The identity relation $I_A$ on set $A$ is the set $\{(a, a) : a \in A\}$. For $A = \{5, 6\}$, $I_A = \{(5, 5), (6, 6)\}$.

Question 9

9. If $A = \{1, 2, 3\}$ and $B = \emptyset$ (empty set), what is the Cartesian product $A \times B$?

A) $\emptyset$

B) $\{(1, 0), (2, 0), (3, 0)\}$

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

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

Answer

Correct Answer: A

If either set $A$ or set $B$ is empty, there are no ordered pairs that can be formed. Therefore, $A \times B = \emptyset$.

Question 10

10. For a relation $R$ from $A$ to $B$, the set $B$ is called the:

A) Domain

B) Range

C) Codomain

D) Universal Set

Answer

Correct Answer: C

For any relation $R$ from $A$ to $B$, the set $B$ is always called the codomain of the relation. Note that the range is a subset of the codomain.

Accepted Answer

Correct Answer: D

The range of a relation is the set of all second components of the ordered pairs. Range = $\{4, 5\}$.

Grade 11 : Mathematics - Relations (Set 1) | Concept of Maths