Section A — MCQ (Single Correct)
Question 1
If $A = \{1, 2\}$ and $B = \{3, 4\}$, then the total number of subsets of the Cartesian product $A \times B$ is:
A
$4$
B
$8$
C
$16$
D
$32$
Question 2
A relation $R$ is defined on the set of natural numbers $\mathbb{N}$ by $R = \{(a, b) : a + 3b = 12\}$. The domain of this relation is:
A
$\{3, 6, 9\}$
B
$\{1, 2, 3\}$
C
$\{2, 4, 6\}$
D
$\mathbb{N}$
Question 3
The relation $R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)\}$ defined on the set $A = \{1, 2, 3\}$ is:
A
An identity relation
B
An equivalence relation
C
A partial order relation
D
Transitive but not symmetric
Question 4
Let $R$ be a relation on the set of real numbers $\mathbb{R}$ defined by $aRb \iff |a - b| \le 1$. This relation is:
A
Reflexive and Symmetric but not Transitive
B
Reflexive and Transitive but not Symmetric
C
An Equivalence relation
D
A Partial Order relation
Question 5
The number of reflexive relations that can be defined on a set with 3 elements is:
A
$8$
B
$64$
C
$512$
D
$256$
Question 6
If $R = \{(1, 2), (3, 4)\}$, then its inverse relation $R^{-1}$ is given by:
A
$\{(2, 1), (4, 3)\}$
B
$\{(1, 2), (3, 4)\}$
C
$\{(2, 1), (3, 4)\}$
D
$\emptyset$
Question 7
On the set of all human beings, the relation ``is a brother of'' is:
A
Symmetric
B
Transitive
C
Reflexive
D
An Equivalence relation
Question 8
The total number of elements inside the identity relation of a set with 10 elements is:
A
$100$
B
$10$
C
$2^{10}$
D
$0$
Question 9
If $R = \{(1, a), (2, b)\}$ and $S = \{(a, 5), (b, 6)\}$, then the composition relation $S \circ R$ is:
A
$\{(1, 5), (2, 6)\}$
B
$\{(a, 1), (b, 2)\}$
C
$\{(5, 1), (6, 2)\}$
D
$\emptyset$
Question 10
A relation $R$ on a set $A$ is anti-symmetric if and only if:
A
$(a, b) \in R \implies (b, a) \notin R$
B
$(a, b) \in R$ and $(b, a) \in R \implies a = b$
C
$(a, a) \in R$ for all $a$
D
It is an empty relation
Section B — Integer Type
Question 11 — Integer answer
Find the total number of distinct equivalence relations that can be formed on a set containing exactly 3 elements.
Question 12 — Integer answer
If $|A| = 2$ and $|B| = 3$, find the total number of distinct relations that can be defined from $A$ to $B$.
Question 13 — Integer answer
Let $R = \{(x, y) \in \mathbb{N} \times \mathbb{N} : x^2 - y^2 = 5\}$. Find the cardinality of this relation set.
Section C — Assertion & Reasoning
Question 14 — Assertion / Reason
Assertion (A): The identity relation $\mathcal{I}_A$ on any non-empty set $A$ is always an equivalence relation.Reason (R): The identity relation satisfies the reflexive, symmetric, and transitive properties simultaneously.
A
Both A and R are true and R is the correct explanation of A
B
Both A and R are true but R is NOT the correct explanation of A
C
A is true but R is false
D
A is false but R is true
Solution: Both A and R are true and R is the correct explanation.
Question 15 — Assertion / Reason
Assertion (A): If $A \times B = \emptyset$, then both sets $A$ and $B$ must be empty sets simultaneously.Reason (R): The Cartesian product is empty if at least one of the component sets is empty ($A = \emptyset$ or $B = \emptyset$).
A
Both A and R are true and R is the correct explanation of A
B
Both A and R are true but R is NOT the correct explanation of A
C
A is true but R is false
D
A is false but R is true
Solution: A is false but R is true.