Completing Statements MCQs for Sub-Topics of Topic 9: Sets, Relations & Functions

Question 1. A well-defined collection of distinct objects is called a ________.

(A) Relation

(B) Function

(C) Set

(D) Collection

Question 2. The symbol '$\in$' is used to denote that an element ________ to a set.

(A) is a subset of

(B) is not an element of

(C) belongs

(D) is equal

Question 3. Listing all the elements of a set, separated by commas and enclosed within curly braces $\{ \}$, is known as the ________ form.

(A) Set-builder

(B) Roster

(C) Descriptive

(D) Symbolic

Question 4. Describing a set by stating a property that its elements satisfy is the ________ form.

(A) Roster

(B) Tabular

(C) Set-builder

(D) Enumeration

Question 5. The set of all natural numbers is denoted by the symbol ________.

(A) $\mathbb{Z}$

(B) $\mathbb{Q}$

(C) $\mathbb{N}$

(D) $\mathbb{R}$

Question 6. The symbol $\mathbb{R}$ represents the set of all ________ numbers.

(A) Rational

(B) Integer

(C) Real

(D) Complex

Question 7. A pair of elements $(a, b)$ where the order of elements is important is called an ________.

(A) Unordered pair

(B) Interval

(C) Ordered pair

(D) Element

Question 8. Two ordered pairs $(a, b)$ and $(c, d)$ are equal if and only if $a=c$ and ________.

(A) $a=d$

(B) $b=c$

(C) $b=d$

(D) $a+b = c+d$

Question 9. The Cartesian product of two sets A and B, denoted by $A \times B$, is the set of all ________ $(a, b)$ where $a \in A$ and $b \in B$.

(A) Unordered pairs

(B) Elements

(C) Ordered pairs

(D) Relations

Question 10. If set A has $m$ elements and set B has $n$ elements, then the Cartesian product $A \times B$ has ________ elements.

(A) $m+n$

(B) $m-n$

(C) $m \times n$

(D) $m^n$

Question 1. A set which contains no elements is called an ________ set.

(A) Infinite

(B) Singleton

(C) Empty

(D) Universal

Question 2. A set which consists of a definite number of elements is called a ________ set.

(A) Infinite

(B) Finite

(C) Universal

(D) Power

Question 3. A set which is not finite is called an ________ set.

(A) Empty

(B) Singleton

(C) Equivalent

(D) Infinite

Question 4. A set containing only one element is called a ________ set.

(A) Null

(B) Doubleton

(C) Singleton

(D) Finite

Question 5. Two sets A and B are said to be ________ if they have exactly the same elements.

(A) Equivalent

(B) Disjoint

(C) Equal

(D) Subsets

Question 6. Two finite sets A and B are said to be ________ if they have the same number of elements, i.e., $n(A) = n(B)$.

(A) Equal

(B) Equivalent

(C) Disjoint

(D) Comparable

Question 7. The cardinal number of a finite set A, denoted by $n(A)$, is the ________ of distinct elements in A.

(A) Sum

(B) Product

(C) Number

(D) Type

Question 8. A set A is a ________ of a set B if every element of A is also an element of B, denoted by $A \subseteq B$.

(A) Superset

(B) Subset

(C) Power set

(D) Universal set

Question 9. The collection of all subsets of a set A is called the ________ of A, denoted by $P(A)$.

(A) Superset

(B) Subset

(C) Power set

(D) Universal set

Question 10. In a particular context, a basic set which includes all sets under consideration is called the ________ set, denoted by U.

(A) Subset

(B) Power

(C) Empty

(D) Universal

Question 1. If A is a subset of B and $A \neq B$, then A is a ________ subset of B, denoted by $A \subset B$.

(A) Proper

(B) Improper

(C) Equal

(D) Superset

Question 2. A ________ R from a set A to a set B is a subset of the Cartesian product $A \times B$.

(A) Function

(B) Operation

(C) Relation

(D) Mapping

Question 3. If $(a, b) \in R$ for a relation R, we say that $a$ is related to $b$ under the relation R, and we write this as ________.

(A) $a \in b$

(B) $a R b$

(C) $R(a) = b$

(D) $a \subseteq b$

Question 4. The set of all first components of the ordered pairs in a relation R from a set A to a set B is called the ________ of R.

(A) Range

(B) Codomain

(C) Domain

(D) Image

Question 5. The set of all second components of the ordered pairs in a relation R from a set A to a set B is called the ________ of R.

(A) Domain

(B) Range

(C) Codomain

(D) Graph

Question 6. For a relation R from set A to set B, the set B is called the ________ of R.

(A) Domain

(B) Range

(C) Codomain

(D) Image

Question 7. The visual representation of a relation using ovals for sets and arrows to show related elements is called an ________.

(A) Venn Diagram

(B) Bar Graph

(C) Arrow Diagram

(D) Scatter Plot

Question 8. The number of relations from a set A with $m$ elements to a set B with $n$ elements is equal to the number of subsets of $A \times B$, which is ________.

(A) $m \times n$

(B) $m+n$

(C) $2^{m+n}$

(D) $2^{mn}$

Question 9. If $A=\{1, 2\}$ and $R$ is a relation on A, then R is a subset of ________.

(A) A

(B) $A \cup A$

(C) $A \times A$

(D) $A \cap A$

Question 10. The relation R on set A defined by $R = \{(a, a) : a \in A\}$ is called the ________ relation on A.

(A) Empty

(B) Universal

(C) Identity

(D) Reflexive

Question 1. A relation R on a set A is ________ if $(a, a) \in R$ for every $a \in A$.

(A) Symmetric

(B) Transitive

(C) Reflexive

(D) Identity

Question 2. A relation R on a set A is ________ if for every $(a, b) \in R$, it is implied that $(b, a) \in R$.

(A) Reflexive

(B) Symmetric

(C) Transitive

(D) Anti-symmetric

Question 3. A relation R on a set A is ________ if for every $(a, b) \in R$ and $(b, c) \in R$, it is implied that $(a, c) \in R$.

(A) Reflexive

(B) Symmetric

(C) Transitive

(D) Equivalence

Question 4. A relation which is reflexive, symmetric, and transitive is called an ________ relation.

(A) Identity

(B) Universal

(C) Equivalence

(D) Partial order

Question 5. The relation R on a non-empty set A defined by $R = A \times A$ is called the ________ relation on A.

(A) Empty

(B) Identity

(C) Universal

(D) Reflexive

Question 6. The relation R on a set A defined by $R = \phi$ is called the ________ relation on A.

(A) Empty

(B) Universal

(C) Identity

(D) Reflexive

Question 7. For an equivalence relation R on a set A, the set of all elements in A that are related to a specific element $a \in A$ is called the ________ of $a$.

(A) Domain

(B) Range

(C) Equivalence class

(D) Image

Question 8. The relation "is equal to" on any set is always a(n) ________ relation.

(A) Reflexive and Symmetric only

(B) Symmetric and Transitive only

(C) Equivalence

(D) Universal

Question 9. If a relation R on a set A is symmetric, then whenever $(a, b) \in R$, ________ must also be in R.

(A) $(a, a)$

(B) $(b, b)$

(C) $(b, a)$

(D) $(a, c)$ for some c

Question 10. For an equivalence relation R on a set A, the equivalence classes form a ________ of the set A.

(A) Subset

(B) Partition

(C) Superset

(D) Union

Question 1. The union of two sets A and B, denoted by $A \cup B$, is the set of all elements which are in A or in B or in ________.

(A) A only

(B) B only

(C) Both A and B

(D) Neither A nor B

Question 2. The intersection of two sets A and B, denoted by $A \cap B$, is the set of all elements which are common to ________ A and B.

(A) Union of

(B) Either

(C) Only

(D) Both

Question 3. The difference of set A and set B, denoted by $A - B$, is the set of elements which are in A but not in ________.

(A) A

(B) B

(C) $A \cap B$

(D) $A \cup B$

Question 4. The complement of a set A, denoted by $A'$, with respect to a universal set U is the set of all elements in U that are not in ________.

(A) U

(B) $\phi$

(C) A

(D) A'

Question 5. In a Venn diagram, the region where two circles representing sets A and B overlap shows the ________ of A and B.

(A) Union

(B) Difference

(C) Complement

(D) Intersection

Question 6. The shaded region in a Venn diagram that covers both circles A and B represents the ________ of A and B.

(A) Intersection

(B) Union

(C) Difference

(D) Complement

Question 7. If A and B are disjoint sets, their intersection $A \cap B$ is the ________ set.

(A) Universal

(B) Empty

(C) Singleton

(D) Finite

Question 8. For any set A, the union of A with the empty set is A, i.e., $A \cup \phi = $ ________.

(A) $\phi$

(B) U

(C) A

(D) A'

Question 9. For any set A, the intersection of A with its complement is the empty set, i.e., $A \cap A' = $ ________.

(A) A

(B) U

(C) $\phi$

(D) A'

Question 10. The region outside the circle representing set A, within the universal rectangle, represents the ________ of A.

(A) Union

(B) Intersection

(C) Difference

(D) Complement

Question 1. The property $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ is called the ________ law.

(A) Commutative

(B) Associative

(C) Distributive

(D) Identity

Question 2. De Morgan's Law states that $(A \cup B)' = $ ________.

(A) $A' \cup B'$

(B) $A \cap B$

(C) $A' \cap B'$

(D) $(A \cap B)'$

Question 3. According to De Morgan's Law, $(A \cap B)' = $ ________.

(A) $A' \cup B'$

(B) $A \cup B$

(C) $A' \cap B'$

(D) $(A \cup B)'$

Question 4. For two finite sets A and B, the formula for the cardinality of their union is $n(A \cup B) = n(A) + n(B) -$ ________.

(A) $n(A \cup B)$

(B) $n(A \cap B)$

(C) $n(A - B)$

(D) $n(B - A)$

Question 5. If A and B are disjoint finite sets, then $n(A \cup B) = $ ________.

(A) $n(A) + n(B) - n(A \cap B)$

(B) $n(A) + n(B)$

(C) $n(A \cap B)$

(D) $n(A) \times n(B)$

Question 6. The principle of inclusion-exclusion is used to calculate the ________ of the union of sets.

(A) Intersection

(B) Difference

(C) Complement

(D) Cardinality

Question 7. For any finite set A, the number of elements in the complement of A, $n(A')$, is given by $n(U) -$ ________, where U is the universal set.

(A) $n(U)$

(B) $n(A)$

(C) $n(A')$

(D) $n(\phi)$

Question 8. The cardinality of the symmetric difference of two finite sets A and B, $n(A \Delta B)$, is equal to $n(A \cup B) -$ ________.

(A) $n(A) + n(B)$

(B) $n(A \cap B)$

(C) $n(A - B)$

(D) $n(B - A)$

Question 9. If $n(A) = 20$, $n(B) = 30$, and $n(A \cup B) = 40$, then $n(A \cap B) =$ ________.

(A) 50

(B) 10

(C) 20

(D) 30

Question 10. If $n(U) = 80$, $n(A) = 50$, and $n(B) = 40$, and $n(A \cap B) = 20$, then $n((A \cup B)') =$ ________.

(A) 10

(B) 20

(C) 30

(D) 40

Question 1. A function is a special type of relation where every element in the domain set is mapped to ________ in the codomain set.

(A) At least one element

(B) Exactly one element

(C) At most one element

(D) More than one element

Question 2. If $f: A \to B$ is a function, the set A is called the ________ of the function.

(A) Range

(B) Codomain

(C) Domain

(D) Image

Question 3. If $f: A \to B$ is a function, the set B is called the ________ of the function.

(A) Domain

(B) Range

(C) Codomain

(D) Pre-image

Question 4. The set of all images of the elements of the domain under a function $f$ is called the ________ of $f$.

(A) Domain

(B) Codomain

(C) Range

(D) Graph

Question 5. The range of a function is always a ________ of its codomain.

(A) Superset

(B) Proper subset

(C) Subset

(D) Union

Question 6. If $f(x) = 2x$, and the domain is $\{1, 2, 3\}$, the range of the function is ________.

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

(B) $\{2, 4, 6\}$

(C) $\{1, 2, 3, 4, 5, 6\}$

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

Question 7. For a function $f: A \to B$, if $(a, b) \in f$, then $b$ is called the image of $a$, and $a$ is called the ________ of $b$ under $f$.

(A) Domain element

(B) Codomain element

(C) Image

(D) Pre-image

Question 8. The domain of the real function $f(x) = \frac{1}{x}$ is the set of all real numbers except ________.

(A) 1

(B) -1

(C) 0

(D) Any positive number

Question 9. The domain of the real function $f(x) = \sqrt{x}$ is the set of all ________ real numbers.

(A) Positive

(B) Negative

(C) Non-negative

(D) Non-positive

Question 10. A relation that satisfies the condition that each element in the domain is associated with exactly one element in the codomain is a ________.

(A) Set

(B) Function

(C) Binary operation

(D) Relation only

Question 1. A function $f: A \to B$ is said to be ________ (or injective) if the images of distinct elements of A under f are distinct.

(A) Onto

(B) Many-to-one

(C) Bijective

(D) One-to-one

Question 2. A function $f: A \to B$ is said to be ________ (or surjective) if every element in the codomain B is the image of at least one element in the domain A.

(A) One-to-one

(B) Into

(C) Bijective

(D) Onto

Question 3. A function that is both one-to-one and onto is called a ________ function.

(A) Many-to-one

(B) Into

(C) Bijective

(D) Identity

Question 4. A function $f: A \to B$ is said to be ________ if two or more distinct elements of A have the same image in B.

(A) One-to-one

(B) Many-to-one

(C) Onto

(D) Bijective

Question 5. A function $f: A \to B$ is said to be ________ if there exists at least one element in the codomain B which is not the image of any element in the domain A.

(A) Onto

(B) Into

(C) Bijective

(D) One-to-one

Question 6. If $f: A \to B$ is a one-to-one function, then for any $x_1, x_2 \in A$, $f(x_1) = f(x_2)$ implies ________.

(A) $x_1 \neq x_2$

(B) $f(x_1) \neq f(x_2)$

(C) $x_1 = x_2$

(D) $f(x_1)$ and $f(x_2)$ are in B

Question 7. If the range of a function $f: A \to B$ is equal to its codomain B, the function is ________.

(A) Injective

(B) Surjective

(C) Bijective

(D) Into

Question 8. If $n(A) < n(B)$ for finite sets A and B, a function $f: A \to B$ cannot be ________.

(A) One-to-one

(B) Onto

(C) Into

(D) Many-to-one

Question 9. If $n(A) > n(B)$ for finite sets A and B, a function $f: A \to B$ cannot be ________.

(A) One-to-one

(B) Onto

(C) Many-to-one

(D) Into

Question 10. A function $f: A \to B$ is bijective if and only if for every element $y \in B$, there is exactly one element $x \in A$ such that $f(x) = $ ________.

(A) y

(B) x

(C) f(y)

(D) f(x)

Question 1. A function $f: A \to B$ is called a real function if both the domain A and the codomain B are subsets of ________.

(A) $\mathbb{N}$

(B) $\mathbb{Z}$

(C) $\mathbb{Q}$

(D) $\mathbb{R}$

Question 2. To find the domain of a real function involving a fraction, we must exclude the values of $x$ that make the ________ zero.

(A) Numerator

(B) Denominator

(C) Function value

(D) Variable

Question 3. To find the domain of a real function involving an even root (like square root), the expression under the root must be ________.

(A) Negative

(B) Positive

(C) Zero

(D) Non-negative

Question 4. The graph of a real function $y = f(x)$ is the set of all points $(x, y)$ such that $x$ is in the domain of $f$ and $y = $ ________.

(A) x

(B) f(x)

(C) y

(D) a constant

Question 5. The vertical line test helps determine if a graph represents a ________.

(A) Relation

(B) Function

(C) One-to-one function

(D) Onto function

Question 6. The horizontal line test helps determine if a graph represents a ________ function.

(A) Onto

(B) Many-to-one

(C) One-to-one

(D) Constant

Question 7. The graph of the identity function $f(x) = x$ is a straight line passing through the origin with a slope of ________.

(A) 0

(B) 1

(C) -1

(D) Undefined

Question 8. The range of the modulus function $f(x) = |x|$ is the set of all ________ real numbers.

(A) Positive

(B) Negative

(C) Non-negative

(D) Integer

Question 9. The graph of a constant function $f(x) = c$ is a ________ line.

(A) Vertical

(B) Horizontal

(C) Slanted

(D) Curved

Question 10. The domain of the sine function $f(x) = \sin x$ is ________.

(A) $[0, 2\pi]$

(B) $[-1, 1]$

(C) $\mathbb{R}$

(D) Integers

Question 1. The sum of two functions $f$ and $g$, denoted by $f+g$, is defined as $(f+g)(x) = $ ________.

(A) $f(g(x))$

(B) $f(x) \times g(x)$

(C) $f(x) + g(x)$

(D) $f(x) - g(x)$

Question 2. The product of two functions $f$ and $g$, denoted by $fg$, is defined as $(fg)(x) = $ ________.

(A) $f(g(x))$

(B) $f(x) \times g(x)$

(C) $f(x) + g(x)$

(D) $f(x) / g(x)$

Question 3. The quotient of two functions $f$ and $g$, denoted by $f/g$, is defined as $(f/g)(x) = f(x)/g(x)$, provided ________.

(A) $f(x) \neq 0$

(B) $g(x) \neq 0$

(C) $x \neq 0$

(D) $f(x) + g(x) \neq 0$

Question 4. The domain of $(f+g)(x)$ is the intersection of the domain of $f$ and the domain of $g$, denoted by $D_f \cap$ ________.

(A) $D_{f+g}$

(B) $D_f \cup D_g$

(C) $D_g$

(D) $D_f \times D_g$

Question 5. The composition of functions $f$ and $g$, denoted by $f \circ g$, is defined as $(f \circ g)(x) = $ ________.

(A) $f(x)g(x)$

(B) $f(x)+g(x)$

(C) $f(g(x))$

(D) $g(f(x))$

Question 6. The composition $f \circ g$ is defined if the range of $g$ is a subset of the ________ of $f$.

(A) Range

(B) Codomain

(C) Domain

(D) Graph

Question 7. If $f(x) = x+1$ and $g(x) = x^2$, then $(f \circ g)(x) = $ ________.

(A) $x^2+1$

(B) $(x+1)^2$

(C) $x(x+1)^2$

(D) $x^2(x+1)$

Question 8. If $f(x) = x+1$ and $g(x) = x^2$, then $(g \circ f)(x) = $ ________.

(A) $x^2+1$

(B) $(x+1)^2$

(C) $x(x+1)^2$

(D) $x^2(x+1)$

Question 9. The operation of composition of functions is generally not ________.

(A) Associative

(B) Commutative

(C) Defined

(D) Distributive

Question 10. If $(f \circ g)(x) = x$ and $(g \circ f)(x) = x$ for all valid $x$, then $g$ is the ________ of $f$, and $f$ is the inverse of $g$.

(A) Domain

(B) Range

(C) Codomain

(D) Inverse

Question 1. A function $f: X \to Y$ is invertible if there exists a function $g: Y \to X$ such that $f \circ g = I_Y$ and ________.

(A) $g \circ f = I_Y$

(B) $g \circ f = I_X$

(C) $f \circ g = g \circ f$

(D) $f(x) = g(x)$

Question 2. A function is invertible if and only if it is ________.

(A) Injective

(B) Surjective

(C) Bijective

(D) Constant

Question 3. If a function is invertible, its inverse is ________.

(A) Not unique

(B) Unique

(C) Always the identity function

(D) Always the zero function

Question 4. To find the inverse of a function $y = f(x)$, we solve the equation for $x$ in terms of $y$, and then usually interchange $x$ and ________.

(A) f(x)

(B) y

(C) the function name

(D) 0

Question 5. A binary operation $*$ on a set S is a function from $S \times S$ to ________.

(A) $S \times S$

(B) S

(C) $\mathbb{R}$

(D) $\mathbb{Z}$

Question 6. A set S is said to be closed under a binary operation $*$ if for all $a, b \in S$, $a * b$ is also an element of ________.

(A) Another set

(B) S

(C) The empty set

(D) The universal set

Question 7. A binary operation $*$ on a set S is commutative if $a * b = $ ________ for all $a, b \in S$.

(A) $a * a$

(B) $b * b$

(C) $b * a$

(D) $(a*b)*c$

Question 8. A binary operation $*$ on a set S is associative if $(a * b) * c = $ ________ for all $a, b, c \in S$.

(A) $a * (c * b)$

(B) $c * (b * a)$

(C) $a * (b * c)$

(D) $(b * c) * a$

Question 9. An element $e \in S$ is an identity element for $*$ if $a * e = a$ and $e * a = a$ for all ________.

(A) $e \in S$

(B) $a \in S$

(C) $a \in S$ and $e \in S$

(D) Some $a \in S$

Question 10. For a binary operation $*$ on S with identity element $e$, an element $b$ is the inverse of $a$ if $a * b = e$ and ________.

(A) $b * a = a$

(B) $b * a = b$

(C) $b * a = e$

(D) $a * b = b * a$