Assertion-Reason MCQs for Sub-Topics of Topic 9: Sets, Relations & Functions

Question 1.

Assertion (A): The collection of all difficult problems in a mathematics textbook is not a set.

Reason (R): A set is a well-defined collection of objects, where it is clear whether an object belongs to the collection or not.

(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.

Question 2.

Assertion (A): The roster form of the set $A = \{x : x \text{ is a positive integer and } x^2 < 10\}$ is $\{1, 2, 3\}$.

Reason (R): The set-builder form describes the property that the elements must satisfy.

(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.

Question 3.

Assertion (A): Every integer is a real number.

Reason (R): The set of integers $\mathbb{Z}$ is a subset of the set of real numbers $\mathbb{R}$.

(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.

Question 4.

Assertion (A): The ordered pair $(2, 3)$ is equal to the ordered pair $(3, 2)$.

Reason (R): In an ordered pair, the order of the elements is significant.

(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.

Question 5.

Assertion (A): If $A=\{a\}$ and $B=\{1, 2\}$, then $A \times B = \{(a, 1), (a, 2)\}$.

Reason (R): The number of elements in the Cartesian product $A \times B$ is the product of the number of elements in A and B.

(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.

Question 1.

Assertion (A): The set of all rivers in India is a finite set.

Reason (R): A finite set is a set which contains a definite number of elements.

(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.

Question 2.

Assertion (A): The sets $A = \{1, 2, 3\}$ and $B = \{a, b, c\}$ are equal sets.

Reason (R): Two sets A and B are equal if they have the same number of elements.

(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.

Question 3.

Assertion (A): The empty set $\phi$ is a subset of every set.

Reason (R): There is no element in $\phi$ that is not in any other set.

(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.

Question 4.

Assertion (A): If a set A has 3 elements, then its power set $P(A)$ has 8 elements.

Reason (R): The number of elements in the power set of a set with n elements is $2^n$.

(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.

Question 5.

Assertion (A): The interval $(2, 5]$ represents the set $\{x \in \mathbb{R} : 2 \leq x \leq 5\}$.

Reason (R): In the interval notation $(a, b]$, the endpoint 'a' is excluded and the endpoint 'b' is included.

(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.

Question 1.

Assertion (A): If $A \subset B$, then $A \subseteq B$.

Reason (R): A proper subset is also a subset.

(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.

Question 2.

Assertion (A): Every subset of the Cartesian product $A \times B$ is a relation from A to B.

Reason (R): A relation from set A to set B is defined as a subset of $A \times B$.

(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.

Question 3.

Assertion (A): Let $A=\{1, 2\}$, $B=\{a, b\}$, and $R = \{(1, a), (2, a)\}$. The domain of R is $\{1, 2\}$.

Reason (R): The domain of a relation R is the set of all second components of the ordered pairs in R.

(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.

Question 4.

Assertion (A): Let $A=\{1, 2\}$, $B=\{a, b\}$, and $R = \{(1, a), (2, a)\}$. The range of R is $\{a\}$.

Reason (R): The range of a relation R is the set of all first components of the ordered pairs in R.

(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.

Question 5.

Assertion (A): If $A=\{1, 2\}$ and $B=\{a\}$, the number of relations from A to B is $2^{1 \times 2} = 4$.

Reason (R): The number of relations from a set A with m elements to a set B with n elements is $2^{mn}$.

(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.

Question 1.

Assertion (A): The identity relation on a non-empty set A is always reflexive.

Reason (R): A relation R on set A is reflexive if $(a, a) \in R$ for all $a \in A$.

(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.

Question 2.

Assertion (A): The relation R = $\{(1, 2), (2, 1)\}$ on the set $A = \{1, 2, 3\}$ is symmetric.

Reason (R): A relation R is symmetric if for every $(a, b) \in R$, $(b, a) \in R$.

(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.

Question 3.

Assertion (A): The relation R on $\mathbb{Z}$ defined by $a R b$ if $a \leq b$ is transitive.

Reason (R): If $a \leq b$ and $b \leq c$, then $a \leq c$ for all integers $a, b, c$.

(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.

Question 4.

Assertion (A): An equivalence relation partitions the set into disjoint equivalence classes.

Reason (R): Every element of the set belongs to exactly one equivalence class.

(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.

Question 5.

Assertion (A): The relation "is perpendicular to" on the set of all lines in a plane is an equivalence relation.

Reason (R): An equivalence relation is reflexive, symmetric, and transitive.

(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.

Question 1.

Assertion (A): For any two sets A and B, $A \cup B = B \cup A$.

Reason (R): The union operation is commutative.

(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.

Question 2.

Assertion (A): If $A = \{1, 2, 3\}$ and $B = \{3, 4, 5\}$, then $A \cap B = \{3\}$.

Reason (R): The intersection of two sets A and B contains all elements that are in A or in B or in both.

(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.

Question 3.

Assertion (A): For any set A, $A - \phi = A$.

Reason (R): The difference $A - B$ contains elements that are in A but not in B.

(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.

Question 4.

Assertion (A): If U is the universal set and A is any set, then $A \cup A' = U$.

Reason (R): The complement $A'$ contains all elements in U that are not in A.

(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.

Question 5.

Assertion (A): In a Venn diagram, the intersection of two circles represents the union of the two sets.

Reason (R): The union of two sets A and B contains elements that are in A and B.

(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.

Question 1.

Assertion (A): For any sets A, B, C, $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$.

Reason (R): Union distributes over intersection.

(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.

Question 2.

Assertion (A): According to De Morgan's Laws, $(A \cap B)' = A' \cap B'$.

Reason (R): The complement of the intersection of two sets is the union of their complements.

(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.

Question 3.

Assertion (A): For two finite sets A and B, $n(A \cup B) = n(A) + n(B) - n(A \cap B)$.

Reason (R): Elements common to both A and B are counted twice when $n(A)$ and $n(B)$ are added separately, so they are subtracted once.

(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.

Question 4.

Assertion (A): In a class of 40 students, 25 like Math and 20 like Science. If 5 like neither, then 10 students like both.

Reason (R): The number of students who like at least one subject is $n(\text{Math} \cup \text{Science}) = n(\text{Math}) + n(\text{Science}) - n(\text{Math} \cap \text{Science})$.

(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.

Question 5.

Assertion (A): For any set A, $A \cap U = A$, where U is the universal set.

Reason (R): The intersection of a set with the universal set gives the universal set.

(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.

Question 1.

Assertion (A): The relation $R = \{(1, a), (2, b), (3, c)\}$ from $A=\{1, 2, 3\}$ to $B=\{a, b, c, d\}$ is a function.

Reason (R): A relation from A to B is a function if every element of A has a unique image in B.

(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.

Question 2.

Assertion (A): For the function $f: \{1, 2, 3\} \to \{a, b, c\}$ defined by $f = \{(1, a), (2, a), (3, b)\}$, the range is $\{a, b, c\}$.

Reason (R): The range of a function is the set of all actual images of the elements in the domain.

(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.

Question 3.

Assertion (A): For a function $f: A \to B$, the codomain B is always equal to the range of $f$.

Reason (R): The range of a function is always a subset of its codomain.

(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.

Question 4.

Assertion (A): The domain of the real function $f(x) = \sqrt{x+4}$ is $[-4, \infty)$.

Reason (R): The expression under a square root must be non-negative.

(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.

Question 5.

Assertion (A): The relation $R = \{(x, y) : x^2 + y^2 = 1\}$ on $\mathbb{R}$ is a function.

Reason (R): For a relation to be a function, each element in the domain must be related to exactly one element in the codomain.

(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.

Question 1.

Assertion (A): The function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 3x + 1$ is one-to-one.

Reason (R): A function $f$ is one-to-one if $f(x_1) = f(x_2)$ implies $x_1 = x_2$ for all $x_1, x_2$ in the domain.

(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.

Question 2.

Assertion (A): The function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^2$ is an onto function.

Reason (R): A function $f: A \to B$ is onto if for every $y \in B$, there exists at least one $x \in A$ such that $f(x) = y$.

(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.

Question 3.

Assertion (A): A function that is both injective and surjective is called bijective.

Reason (R): A bijective function establishes a one-to-one correspondence between the domain and the codomain.

(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.

Question 4.

Assertion (A): If $n(A) > n(B)$, then there cannot be a one-to-one function from A to B.

Reason (R): In a one-to-one function $f: A \to B$, different elements of A must have different images in B.

(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.

Question 5.

Assertion (A): The function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = \cos x$ is a many-to-one function.

Reason (R): In a many-to-one function, at least two different elements in the domain have the same image in the codomain.

(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.

Question 1.

Assertion (A): The function $f: \mathbb{N} \to \mathbb{R}$ defined by $f(x) = x^2$ is a real function.

Reason (R): A function $f: A \to B$ is a real function if A and B are subsets of $\mathbb{R}$.

(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.

Question 2.

Assertion (A): The domain of the real function $f(x) = \frac{1}{x^2+1}$ is $\mathbb{R}$.

Reason (R): The denominator $x^2+1$ is never zero for any real number x.

(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.

Question 3.

Assertion (A): The range of the modulus function $f(x) = |x|$ is $\mathbb{R}$.

Reason (R): The modulus of any real number is always non-negative.

(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.

Question 4.

Assertion (A): A vertical line can intersect the graph of a function at most once.

Reason (R): This is known as the vertical line test for functions, ensuring each input has only one output.

(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.

Question 5.

Assertion (A): The graph of $f(x) = x^2$ is a parabola symmetric about the y-axis.

Reason (R): For the function $f(x) = x^2$, $f(-x) = (-x)^2 = x^2 = f(x)$, which indicates symmetry about the y-axis.

(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.

Question 1.

Assertion (A): If $f(x) = x+1$ and $g(x) = x-1$, then $(f+g)(x) = 2x$.

Reason (R): The sum of two functions $(f+g)(x)$ is defined as $f(x) + g(x)$.

(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.

Question 2.

Assertion (A): If $f(x) = x^2$ and $g(x) = \sqrt{x}$, the domain of $(f/g)(x)$ is $(0, \infty)$.

Reason (R): The domain of $(f/g)(x)$ is the intersection of the domains of f and g, excluding points where $g(x) = 0$.

(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.

Question 3.

Assertion (A): If $f(x) = 2x$ and $g(x) = x+3$, then $(f \circ g)(x) = 2x+6$.

Reason (R): The composition $(f \circ g)(x)$ is defined as $f(g(x))$.

(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.

Question 4.

Assertion (A): For any two functions f and g, $f \circ g = g \circ f$.

Reason (R): Composition of functions is always commutative.

(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.

Question 5.

Assertion (A): The domain of the composite function $(f \circ g)(x)$ is the domain of $g(x)$ if the range of $g$ is a subset of the domain of $f$.

Reason (R): The composite function $f \circ g$ is defined if and only if the range of $g$ is a subset of the domain of $f$.

(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.

Question 1.

Assertion (A): The function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^2$ is not invertible.

Reason (R): A function is invertible if and only if it is bijective (both one-to-one and onto).

(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.

Question 2.

Assertion (A): The inverse of the function $f(x) = x - 7$ is $f^{-1}(x) = x + 7$.

Reason (R): If $y = f(x)$, the inverse function $f^{-1}(y)$ is found by solving for x in terms of y and then swapping x and y.

(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.

Question 3.

Assertion (A): Subtraction (-) is a binary operation on the set of natural numbers $\mathbb{N}$.

Reason (R): A binary operation $*$ on a set S maps every ordered pair of elements from S to an element in S.

(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.

Question 4.

Assertion (A): The binary operation $*$ on $\mathbb{R}$ defined by $a * b = a + b - ab$ is commutative.

Reason (R): A binary operation $*$ is commutative if $a * b = b * a$ for all $a, b$ in the set.

(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.

Question 5.

Assertion (A): For the binary operation of multiplication ($\times$) on the set of integers $\mathbb{Z}$, the identity element is 1.

Reason (R): An element $e$ is an identity element for $*$ on S if $a * e = a$ and $e * a = a$ for all $a \in S$.

(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.