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: Option A

Explanation: 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) 6B) 32 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: Option B

Explanation: 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?

A) \{1, 2, 3\}B) \{4\}C) \{5\}D) \{4, 5\}

Answer

✓ Correct Answer: Option D

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

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: Option A

Explanation: 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: Option C

Explanation: 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: Option B

Explanation: 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: Option D

Explanation: 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) \emptysetB) \{(1, 0), (2, 0), (3, 0)\}C) \{1, 2, 3\}D) \{(1, \emptyset), (2, \emptyset), (3, \emptyset)\}

Answer

✓ Correct Answer: Option A

Explanation: 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) DomainB) RangeC) CodomainD) Universal Set

Answer

✓ Correct Answer: Option C

Explanation: 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: Option C

Explanation: 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 = $2^6 = 64$.

Grade 11 : Mathematics - Relations (Set 1)

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

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

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

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?

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?

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

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

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

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

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