Negative Questions MCQs for Sub-Topics of Topic 9: Sets, Relations & Functions

Question 1. Which of the following is NOT a well-defined collection of objects, and therefore not a set?

(A) The collection of all continents in the world.

(B) The collection of all prime ministers of India.

(C) The collection of all dangerous animals in the zoo.

(D) The collection of all states in India.

Question 2. Let $A = \{x : x \text{ is an integer and } |x| \leq 2\}$. Which of the following elements does NOT belong to set A?

(A) -2

(B) 0

(C) 2

(D) 3

Question 3. Which of the following is NOT a correct representation in roster form for the set of vowels in the English alphabet?

(A) $\{a, e, i, o, u\}$

(B) $\{A, E, I, O, U\}$

(C) $\{a, i, o, u, e\}$

(D) $\{a, e, i, o, u, y\}$

Question 4. Which of the following is NOT a correct set-builder form for the set $\{3, 6, 9\}$?

(A) $\{x : x = 3n, n \in \mathbb{N}, 1 \leq n \leq 3\}$

(B) $\{x : x \text{ is a positive multiple of 3 and } x \leq 9\}$

(C) $\{x : x \text{ is a multiple of 3 and } 3 \leq x \leq 9\}$

(D) $\{x : x \text{ is a prime number and } x < 10\}$

Question 5. Which of the following notations does NOT represent a standard set of numbers?

(A) $\mathbb{N}$ (Set of natural numbers)

(B) $\mathbb{S}$ (Set of special numbers)

(C) $\mathbb{Z}$ (Set of integers)

(D) $\mathbb{Q}$ (Set of rational numbers)

Question 6. Which of the following statements about ordered pairs is FALSE?

(A) The ordered pair $(a, b)$ is equal to $(b, a)$ for all values of $a$ and $b$.

(B) If $(x, y) = (5, 7)$, then $x=5$ and $y=7$.

(C) The order of elements in an ordered pair is important.

(D) An ordered pair is distinct from a set containing the same elements.

Question 7. If $(p+2, q-1) = (7, 4)$, which of the following statements is NOT true?

(A) $p+2 = 7$

(B) $q-1 = 4$

(C) $p=5$

(D) $q=6$

Question 8. Let $A=\{a\}$ and $B=\{1, 2, 3\}$. Which of the following is NOT an element of the Cartesian product $A \times B$?

(A) $(a, 1)$

(B) $(a, 2)$

(C) $(3, a)$

(D) $(a, 3)$

Question 9. If $n(A)=4$ and $n(B)=5$, which of the following is NOT the number of elements in $A \times B$?

(A) 20

(B) $4 \times 5$

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

(D) 9

Question 10. Let $A = \{1, 2\}$. Which of the following is NOT an element of $A \times A$?

(A) $(1, 1)$

(B) $(2, 2)$

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

(D) $(1, 2)$

Question 1. Which of the following is NOT an empty set?

(A) $\{x : x \in \mathbb{R}, x^2 = -1\}$

(B) $\{x : x \text{ is a student studying in both Class 10 and Class 11}\}$

(C) $\{x : x \in \mathbb{N}, 2 < x < 3\}$

(D) $\{0\}$

Question 2. Which of the following is NOT a finite set?

(A) The set of days in a week.

(B) The set of population of India.

(C) The set of even prime numbers.

(D) The set of all points on a line segment.

Question 3. Which of the following is NOT an infinite set?

(A) The set of all triangles in a plane.

(B) The set of rational numbers.

(C) The set of solutions to $x^2 - 5x + 6 = 0$ in $\mathbb{R}$.

(D) The set of irrational numbers.

Question 4. Which of the following is NOT a singleton set?

(A) $\{x : x \in \mathbb{Z}, x^2 = 0\}$

(B) $\{x : x \in \mathbb{N}, 1 < x < 3\}$

(C) $\{\phi\}$

(D) $\{x : x \in \mathbb{R}, \sqrt{x} = -2\}$

Question 5. Let $A = \{a, b\}$. Which of the following sets is NOT equal to A?

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

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

(C) $\{x : x \text{ is a letter in the word 'ABBA'}\}$

(D) $\{a, b, c\}$

Question 6. Let $A = \{1, 2, 3\}$ and $B = \{a, b\}$. Which of the following statements is NOT true?

(A) A and B are equivalent sets.

(B) $n(A) \neq n(B)$

(C) A is a superset of B.

(D) The number of elements in A is 3.

Question 7. Let $S = \{1, 2, 3\}$. Which of the following is NOT a subset of S?

(A) $\{1\}$

(B) $\phi$

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

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

Question 8. If A is a set, which of the following statements is FALSE?

(A) $\phi \subseteq A$

(B) $A \subseteq A$

(C) Every element of A is a subset of A.

(D) The number of proper subsets of A with $n$ elements is $2^n - 1$.

Question 9. Let $S = \{1, 2\}$. Which of the following is NOT an element of the power set $P(S)$?

(A) $\{1\}$

(B) $\{2\}$

(C) $\phi$

(D) 1

Question 10. Which of the following does NOT represent an interval on the real number line?

(A) $[2, 7)$

(B) $(-\infty, 5]$

(C) $\{x : x \in \mathbb{R}, x^2 < 0\}$

(D) $\mathbb{R}$

Question 1. Let $A = \{1, 2, 3\}$, $B = \{1, 2\}$. Which of the following statements is NOT true?

(A) $B \subset A$

(B) $B \subseteq A$

(C) A is a superset of B.

(D) $A \subseteq B$

Question 2. If $A=\{1, 2\}$ and $B=\{a, b\}$, which of the following is NOT a relation from A to B?

(A) $\phi$

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

(C) $\{(1, a), (2, b), (3, a)\}$

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

Question 3. Let $A = \{1, 2, 3\}$. A relation R on A is defined as $R = \{(x, y) : x \leq y\}$. Which of the following ordered pairs is NOT in R?

(A) $(1, 1)$

(B) $(2, 3)$

(C) $(3, 2)$

(D) $(1, 3)$

Question 4. For the relation $R = \{(1, a), (2, b), (3, a)\}$ from $A=\{1, 2, 3\}$ to $B=\{a, b, c\}$, which of the following is NOT true?

(A) The domain of R is $\{1, 2, 3\}$.

(B) The range of R is $\{a, b\}$.

(C) The codomain of R is $\{a, b, c\}$.

(D) The range of R is equal to the codomain of R.

Question 5. Let $A=\{2, 3\}$ and $B=\{4, 6, 9\}$. A relation R from A to B is "is a factor of". Which of the following is NOT an element of R?

(A) $(2, 4)$

(B) $(2, 6)$

(C) $(3, 9)$

(D) $(3, 4)$

Question 6. If a relation R from A to B contains only ordered pairs where the first element is from A and the second element is from B, which of the following statements is FALSE?

(A) R is a subset of $A \times B$.

(B) The domain of R is a subset of A.

(C) The range of R is a subset of B.

(D) R must be equal to $A \times B$.

Question 7. If $n(A) = p$ and $n(B) = q$, the number of possible relations from A to B is $2^{pq}$. Which of the following is NOT a possible number of relations from A to B if $A=\{1, 2\}, B=\{a, b\}$?

(A) $2^{2 \times 2} = 16$

(B) The number of subsets of $A \times B$.

(C) $2^{2+2} = 16$

(D) $2^4$

Question 8. Let R be a relation on set A. Which of the following is NOT necessarily true?

(A) $R \subseteq A \times A$

(B) Domain of R $\subseteq$ A

(C) Range of R $\subseteq$ A

(D) Domain of R = A

Question 9. Consider the relation "is the wife of" on the set of all people. Which of the following ordered pairs would NOT be in this relation?

(A) (Priya, Rahul) where Priya is Rahul's wife.

(B) (Amit, Simran) where Amit is Simran's husband.

(C) (Kavita, Kavita)

(D) (Suresh, Geeta) where Suresh is Geeta's wife.

Question 10. If a relation R from A to B is the empty set $\phi$, which of the following statements is FALSE?

(A) The domain of R is $\phi$.

(B) The range of R is $\phi$.

(C) R is a subset of $A \times B$.

(D) R contains at least one ordered pair.

Question 1. Which of the following relations on a non-empty set A is NOT always reflexive?

(A) The universal relation $A \times A$

(B) The identity relation $I_A$

(C) The empty relation $\phi$

(D) An equivalence relation on A

Question 2. Let A be the set of all lines in a plane. The relation R on A defined by $l_1 R l_2$ if "$l_1$ is parallel to $l_2$". Which property does R NOT satisfy?

(A) Reflexive

(B) Symmetric

(C) Transitive

(D) It is an equivalence relation.

Question 3. Let A be the set of integers $\mathbb{Z}$. The relation R on A defined by $a R b$ if $a \leq b$. Which property does R NOT satisfy?

(A) Reflexive

(B) Symmetric

(C) Transitive

(D) None of the above (i.e., it satisfies all listed properties)

Question 4. Let $A = \{1, 2, 3\}$. Consider the relation $R = \{(1, 2), (2, 1)\}$. Which property does R NOT satisfy on the set A?

(A) Symmetric

(B) Transitive (vacuously true since there is no chain aRb and bRc)

(C) Reflexive

(D) None of the above (i.e., it satisfies all listed properties)

Question 5. Let R be a relation on $\mathbb{Z}$ defined by $a R b$ if $a b \geq 0$. Which property does R NOT satisfy?

(A) Reflexive (since $a^2 \geq 0$)

(B) Symmetric (since $ab = ba$)

(C) Transitive (e.g., $2 R 3$ (6>=0), $3 R -1$ (-3<0) - not transitive, need counter example $2R3, 3R-1$, $2R-1$ is false; $2R(-3)$ (-6>=0 F), $-3R(-5)$ (15>=0 T), $2R(-5)$ (-10>=0 F) - not transitive, e.g., $2 R (-3)$ is false. Consider $a R b$ and $b R c$. $(2, 0) \in R$, $(0, -3) \in R$, but $(2, -3) \notin R$. So, it is NOT transitive.)

(D) None of the above (i.e., it satisfies all listed properties)

Question 6. Which of the following statements about equivalence relations is FALSE?

(A) Every equivalence relation is reflexive, symmetric, and transitive.

(B) An equivalence relation partitions the set into disjoint equivalence classes.

(C) All elements in an equivalence class are related to each other.

(D) An equivalence relation can relate an element to itself and to other elements in the same class, but never to elements outside its class.

(E) The identity relation on a set is not an equivalence relation.

Question 7. Which of the following is NOT a required property for R to be an equivalence relation?

(A) Reflexivity

(B) Symmetry

(C) Transitivity

(D) Asymmetry

Question 8. Let $A=\{1, 2, 3\}$. Which of the following relations on A is NOT symmetric?

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

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

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

(D) $\phi$

Question 9. Let $A=\{a, b, c\}$. Which of the following relations on A is NOT transitive?

(A) $\{(a, b), (b, c), (a, c)\}$

(B) $\{(a, a)\}$

(C) $\{(a, b), (b, c), (c, a)\}$ (Not transitive)

(D) $\phi$

Question 10. Which of the following is NOT a type of relation discussed in this chapter?

(A) Empty relation

(B) Universal relation

(C) Identity relation

(D) Functional relation

Question 1. Let $A = \{1, 2, 3, 4\}$ and $B = \{3, 4, 5, 6\}$. Which of the following is NOT an element of $A \cup B$?

(A) 1

(B) 3

(C) 5

(D) 7

Question 2. Let $A = \{a, b, c\}$ and $B = \{c, d, e\}$. Which of the following is NOT an element of $A \cap B$?

(A) a

(B) c

(C) d

(D) e

Question 3. Let $A = \{1, 2, 3, 4\}$ and $B = \{3, 4, 5\}$. Which of the following is NOT an element of $A - B$?

(A) 1

(B) 2

(C) 3

(D) 4

Question 4. Let $U = \{1, 2, 3, 4, 5, 6\}$ and $A = \{2, 4, 6\}$. Which of the following is NOT an element of $A'$ (complement of A)?

(A) 1

(B) 3

(C) 5

(D) 6

Question 5. If A and B are two sets, which of the following is NOT a valid set operation?

(A) $A + B$

(B) $A \cup B$

(C) $A \cap B$

(D) $A - B$

Question 6. In a Venn diagram illustrating two sets A and B within a universal set U, which area does NOT represent elements only in A or only in B or in both?

(A) The region where A and B overlap.

(B) The region inside circle A but outside circle B.

(C) The region inside circle B but outside circle A.

(D) The region outside both circles A and B, but inside the rectangle U.

Question 7. If A and B are disjoint sets, which of the following statements is NOT true?

(A) $A \cap B = \phi$

(B) $A \cup B = A + B$ (in terms of cardinality for finite sets)

(C) $A \cap B \neq \phi$

(D) The intersection region in their Venn diagram is empty.

Question 8. For any set A, which of the following identities is FALSE?

(A) $A \cup \phi = A$

(B) $A \cap A = A$

(C) $A \cup A' = A$

(D) $A \cap U = A$

Question 9. Which of the following statements about the complement operation is FALSE?

(A) $(A')' = A$

(B) $U' = \phi$

(C) $\phi' = U$

(D) $A \cup A' = \phi$

Question 10. The shaded region in the Venn diagram below represents $A \cup B$. Which description does NOT correspond to this shaded region?

(A) Elements in A only.

(B) Elements in B only.

(C) Elements in A and B.

(D) Elements outside A and B.

Question 1. Which of the following is NOT a correct statement of set algebra identity?

(A) $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$

(B) $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$

(C) $(A \cup B)' = A' \cup B'$

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

Question 2. Which of the following is NOT a statement of De Morgan's Law?

(A) $(A \cup B)' = A' \cap B'$

(B) $(A \cap B)' = A' \cup B'$

(C) $A - B = A \cap B'$

(D) $A \cup B = A \cap B$

Question 3. For two finite sets A and B, which of the following formulas for $n(A \cup B)$ is INCORRECT?

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

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

(C) $n(A) + n(B)$ (only if A and B are disjoint)

(D) $n(A) + n(B)$ (always)

Question 4. If $n(A) = 18$, $n(B) = 22$, and $n(A \cup B) = 35$, which of the following statements is FALSE?

(A) $n(A \cap B) = 18 + 22 - 35 = 5$

(B) $n(A - B) = 18 - 5 = 13$

(C) $n(B - A) = 22 - 5 = 17$

(D) $n(A \Delta B) = n(A \cup B) - n(A \cap B) = 35 - 5 = 30$

(E) $n(A \cup B) = n(A) + n(B)$

Question 5. In a class of 60 students, 40 like Science and 30 like Math. If 10 like neither, which of the following is FALSE?

(A) Number of students who like at least one subject = $60 - 10 = 50$.

(B) Let S = Science, M = Math. $n(S \cup M) = 50$.

(C) $n(S \cap M) = n(S) + n(M) - n(S \cup M) = 40 + 30 - 50 = 20$.

(D) Number of students who like Science only = $n(S) - n(S \cap M) = 40 - 20 = 20$.

(E) Number of students who like Math only = $n(M) - n(S \cap M) = 30 - 20 = 10$.

(F) The number of students who like at least one subject is $40 + 30 = 70$.

Question 6. For any finite set A, which of the following statements is FALSE?

(A) $n(\phi) = 0$

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

(C) $n(A \cup \phi) = n(A) + n(\phi)$

(D) $n(A) + n(A') = n(U)$

(E) $n(A - A) = n(A)$

Question 7. Which of the following statements is NOT derived from the Principle of Inclusion-Exclusion for three sets A, B, C?

(A) $n(A \cup B \cup C) = n(A)+n(B)+n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)$

(B) $n(\text{only A}) = n(A) - n(A \cap B) - n(A \cap C) + n(A \cap B \cap C)$

(C) $n(\text{only A and B}) = n(A \cap B) - n(A \cap B \cap C)$

(D) $n(\text{none of A, B, C}) = n(U) - n(A) - n(B) - n(C) + n(A \cap B) + n(B \cap C) + n(A \cap C) - n(A \cap B \cap C)$

(E) $n(A \cup B \cup C) = n(A)+n(B)+n(C)$ (always)

Question 8. If $n(A \cup B) = 40$, $n(A) = 25$, $n(B) = 30$, which of the following is NOT the value of $n(A \cap B)$?

(A) $25 + 30 - 40$

(B) 15

(C) 40 - $n(A - B) - n(B - A)$

(D) 10

Question 9. The symmetric difference of two sets A and B is $A \Delta B = (A - B) \cup (B - A)$. Which statement is FALSE?

(A) $A \Delta B = (A \cup B) - (A \cap B)$

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

(C) $n(A \Delta B) = n(A) + n(B) - 2n(A \cap B)$

(D) $A \Delta B = B \Delta A$

Question 10. Which of the following statements about set operations and cardinality is FALSE?

(A) $A \cap A = A$

(B) $A \cup (B \cup C) = (A \cup B) \cup C$

(C) $n(A \times B) = n(A) + n(B)$

(D) $n(A - B) \leq n(A)$

Question 1. Which of the following relations from $A=\{1, 2\}$ to $B=\{a, b\}$ is NOT a function?

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

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

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

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

Question 2. If $f: A \to B$ is a function, which of the following statements is FALSE?

(A) Every element of A has an image in B.

(B) Every element of A has a unique image in B.

(C) The range of $f$ is always equal to the codomain B.

(D) The domain of $f$ is A.

Question 3. Let $f = \{(1, p), (2, q), (3, r)\}$ be a function from $A=\{1, 2, 3\}$ to $B=\{p, q, r, s\}$. Which of the following statements is NOT true?

(A) The domain of $f$ is $\{1, 2, 3\}$.

(B) The range of $f$ is $\{p, q, r\}$.

(C) The codomain of $f$ is $\{p, q, r, s\}$.

(D) The range of $f$ is equal to the codomain of $f$.

Question 4. If $f(x) = x^2$, with domain $\mathbb{R}$, which of the following is NOT in the range of $f$?

(A) 9

(B) 0

(C) -4

(D) 1

Question 5. If $f(x) = \sqrt{x-1}$, which of the following values is NOT in the domain of the real function f?

(A) 1

(B) 5

(C) 0

(D) 10

Question 6. If $f(x) = \frac{1}{x+3}$, which of the following values is NOT in the domain of the real function f?

(A) 0

(B) -3

(C) 5

(D) 10

Question 7. Which of the following relations from $A=\{a, b, c\}$ to $B=\{p, q\}$ IS a function?

(A) $\{(a, p), (b, q), (c, p)\}$

(B) $\{(a, p), (b, q)\}$

(C) $\{(a, p), (a, q), (b, p), (c, q)\}$

(D) $\{(a, p), (b, p), (c, q), (a, q)\}$

Question 8. For the function $f(x) = |x|$, with domain $\mathbb{R}$, which of the following values is NOT in the range of f?

(A) 5

(B) 0

(C) 2.5

(D) -1

Question 9. If a function $f: A \to B$ is defined, which of the following is NOT necessarily true?

(A) Domain of $f$ is A.

(B) Range of $f$ is a subset of B.

(C) Codomain of $f$ is B.

(D) Every element in B has a pre-image in A.

Question 10. Which of the following is NOT a defining characteristic of a function from set A to set B?

(A) Every element in A is related to some element in B.

(B) Each element in A is related to exactly one element in B.

(C) The set of ordered pairs $(x, y)$ where $x \in A, y \in B$ is a subset of $A \times B$.

(D) Every element in B is related to some element in A.

Question 1. Which of the following functions from $\mathbb{R} \to \mathbb{R}$ is NOT one-to-one (injective)?

(A) $f(x) = 5x - 2$

(B) $f(x) = x^3$

(C) $f(x) = x^2$

(D) $f(x) = x + 7$

Question 2. Which of the following functions from $\mathbb{R} \to \mathbb{R}$ is NOT onto (surjective)?

(A) $f(x) = x + 10$

(B) $f(x) = x^3$

(C) $f(x) = \sin x$

(D) $f(x) = -2x + 5$

Question 3. Which of the following functions is NOT bijective?

(A) $f: \mathbb{R} \to \mathbb{R}$, $f(x) = x + 5$

(B) $f: \mathbb{Z} \to \mathbb{Z}$, $f(x) = x^2$

(C) $f: \{1, 2, 3\} \to \{a, b, c\}$, $f = \{(1, a), (2, b), (3, c)\}$

(D) $f: \mathbb{R}^+ \to \mathbb{R}^+$, $f(x) = 1/x$

Question 4. Which of the following functions from $\mathbb{R} \to \mathbb{R}$ is NOT many-to-one?

(A) $f(x) = x^2$

(B) $f(x) = |x|$

(C) $f(x) = \cos x$

(D) $f(x) = 3x - 4$

Question 5. Which of the following functions from $\mathbb{R} \to \mathbb{R}$ is NOT into (i.e., is onto)?

(A) $f(x) = x^2$

(B) $f(x) = |x|$

(C) $f(x) = e^x$ (Range is $(0, \infty)$)

(D) $f(x) = 2x - 1$

Question 6. Let $A = \{1, 2, 3\}$ and $B = \{p, q\}$. Which of the following types of functions from A to B is NOT possible?

(A) Many-to-one

(B) Onto

(C) One-to-one

(D) Into

Question 7. Let $A = \{1, 2\}$ and $B = \{p, q, r\}$. Which of the following types of functions from A to B is NOT possible?

(A) One-to-one

(B) Into

(C) Onto

(D) Many-to-one

Question 8. If $f: A \to B$ is an injective function between finite sets A and B, which of the following is NOT necessarily true?

(A) $n(A) \leq n(B)$

(B) Different elements in A have different images in B.

(C) Every element in B has at most one pre-image in A.

(D) $f$ is also surjective.

Question 9. If $f: A \to B$ is a surjective function between finite sets A and B, which of the following is NOT necessarily true?

(A) $n(A) \geq n(B)$

(B) The range of $f$ is equal to B.

(C) Every element in B has at least one pre-image in A.

(D) $f$ is also injective.

Question 10. Which of the following is NOT a necessary condition for a function $f: A \to B$ to be bijective?

(A) f must be one-to-one.

(B) f must be onto.

(C) $n(A) = n(B)$ (if A and B are finite).

(D) There is a function $g: B \to A$ such that $f \circ g = g \circ f = I_A$.

Question 1. A function $f: A \to B$ is a real function. Which of the following is NOT necessarily true?

(A) A is a subset of $\mathbb{R}$.

(B) B is a subset of $\mathbb{R}$.

(C) The domain of $f$ is $\mathbb{R}$.

(D) The range of $f$ is a subset of $\mathbb{R}$.

Question 2. Which of the following is NOT a common issue when finding the domain of a real function?

(A) Denominator being zero.

(B) Expression under an even root being negative.

(C) Expression under an odd root being negative.

(D) Taking the logarithm of a non-positive number.

Question 3. The domain of the real function $f(x) = \frac{1}{\sqrt{x+2}}$ is: Which of the following values is NOT in the domain?

(A) 0

(B) -1

(C) -2

(D) 5

Question 4. The range of the real function $f(x) = |x-5|$ is: Which of the following values is NOT in the range?

(A) 0

(B) 10

(C) -3

(D) 5.5

Question 5. Which of the following statements about the graph of a real function $y = f(x)$ is FALSE?

(A) The set of x-coordinates of the points on the graph is the domain of the function.

(B) The set of y-coordinates of the points on the graph is the range of the function.

(C) A vertical line intersects the graph at most once.

(D) A horizontal line intersects the graph at most once.

Question 6. Which of the following is NOT a standard real function discussed in typical introductory courses?

(A) Identity function

(B) Constant function

(C) Modulus function

(D) Lucky number function

Question 7. The graph of the function $f(x) = x^3$ passes through which of the following points?

(A) $(0, 0)$

(B) $(1, 1)$

(C) $(-1, -1)$

(D) $(2, 6)$

Question 8. The range of the sine function $f(x) = \sin x$ is $[-1, 1]$. Which of the following values is NOT in the range?

(A) 0.5

(B) -1

(C) 1.5

(D) 0

Question 9. The function $f(x) = \begin{cases} x+1 & , & x < 0 \\ x-1 & , & x \geq 0 \end{cases}$. Which of the following is NOT true about the function?

(A) $f(-2) = -1$

(B) $f(0) = -1$

(C) $f(3) = 2$

(D) The function is continuous at $x=0$.

Question 10. Which of the following graphs does NOT represent a function?

(A) Graph 1

(B) Graph 2

(C) Graph 3

(D) Graph 4

Question 1. Let $f(x) = x^2$ and $g(x) = 3x$. Which of the following is NOT a correct result of the algebraic operations?

(A) $(f+g)(x) = x^2 + 3x$

(B) $(f-g)(x) = x^2 - 3x$

(C) $(fg)(x) = 3x^3$

(D) $(f/g)(x) = \frac{x}{3}$ (for $x \neq 0$)

Question 2. Let $f(x) = \sqrt{x+1}$ and $g(x) = x^2$. The domain of $f$ is $[-1, \infty)$ and the domain of $g$ is $\mathbb{R}$. Which of the following is NOT in the domain of $(f+g)(x)$?

(A) 0

(B) -1

(C) -2

(D) 5

Question 3. Let $f(x) = x-2$ and $g(x) = x+5$. The domain of $(f/g)(x)$ is $\mathbb{R} - \{-5\}$. Which of the following values is NOT in the domain of $(f/g)(x)$?

(A) -5

(B) 2

(C) 0

(D) 10

Question 4. Let $f(x) = x+5$ and $g(x) = x^2$. Which of the following composite function expressions is INCORRECT?

(A) $(f \circ g)(x) = x^2 + 5$

(B) $(g \circ f)(x) = (x+5)^2$

(C) $(f \circ g)(x) = (x+5)^2$

(D) $(g \circ f)(x) = x^2 + 5$

Question 5. The composition of functions $f \circ g$ is defined if: Which condition is NOT required?

(A) The domain of $f$ is non-empty.

(B) The domain of $g$ is non-empty.

(C) The range of $g$ is a subset of the domain of $f$.

(D) The codomain of $g$ is equal to the domain of $f$.

Question 6. Let $f(x) = \sqrt{x}$ (domain $[0, \infty)$) and $g(x) = x-3$ (domain $\mathbb{R}$). Which of the following is NOT in the domain of $(f \circ g)(x) = f(g(x)) = \sqrt{x-3}$?

(A) 3

(B) 5

(C) 0

(D) 10

Question 7. Let $f(x) = x+1$. Which of the following statements about composition is FALSE?

(A) $(f \circ f)(x) = x+2$

(B) $(f \circ f \circ f)(x) = x+3$

(C) $f \circ f$ is associative.

(D) Composition with f is commutative with itself.

Question 8. Let $f(x) = 3x$ and $g(x) = x/3$. Which of the following is FALSE?

(A) $(f \circ g)(x) = x$

(B) $(g \circ f)(x) = x$

(C) $f$ and $g$ are inverses of each other.

(D) $(f \circ g)(x) = (g \circ f)(x)$ always.

(E) $(f \circ g)(x) = 9x$

Question 9. Which statement about the algebra of functions is FALSE?

(A) $(f+g)(x) = f(x) + g(x)$

(B) $(fg)(x) = f(x)g(x)$

(C) $(f/g)(x) = f(x)/g(x)$, provided $g(x) \neq 0$.

(D) The domain of $(f+g)$ is always the union of the domains of f and g.

Question 10. Let $f(x) = x^2$ and $g(x) = x+1$. Which of the following is NOT true?

(A) $(f+g)(1) = 3$

(B) $(f \circ g)(1) = 4$

(C) $(g \circ f)(1) = 2$

(D) $(f-g)(0) = -1$

(E) $(fg)(-1) = 0$

(F) $(f/g)(0) = 0$

(G) $(f/g)(-1) = \text{undefined}$

Question 1. A function $f: X \to Y$ is invertible if and only if it satisfies which of the following conditions?

(A) f is one-to-one.

(B) f is onto.

(C) f is bijective.

(D) The range of f is a proper subset of the codomain Y.

Question 2. If a function $f$ is invertible, its inverse function $f^{-1}$ has certain properties. Which of the following is FALSE about $f^{-1}$?

(A) $f^{-1}$ is unique.

(B) The domain of $f^{-1}$ is the codomain of $f$.

(C) The range of $f^{-1}$ is the domain of $f$.

(D) $f \circ f^{-1}$ is the identity function on the domain of $f^{-1}$, and $f^{-1} \circ f$ is the identity function on the domain of $f$.

(E) The inverse function is always defined for any function.

Question 3. Which of the following real functions is NOT invertible over the domain $\mathbb{R}$ and codomain $\mathbb{R}$?

(A) $f(x) = x + 1$

(B) $f(x) = 5x$

(C) $f(x) = x^2$

(D) $f(x) = -x + 2$

Question 4. If $f(x) = 3x - 2$, which of the following statements about its inverse is FALSE?

(A) The inverse function exists.

(B) The inverse function is $f^{-1}(x) = \frac{x+2}{3}$.

(C) $f \circ f^{-1}(x) = x$ for all $x$ in the domain of $f^{-1}$.

(D) The graph of $f^{-1}(x)$ is the same as the graph of $f(x)$.

Question 5. A binary operation $*$ on a set S is a function from: Which is the INCORRECT mapping?

(A) $S \times S \to S$

(B) $(a, b) \mapsto a * b$

(C) Ordered pairs of elements from S to a unique element in S.

(D) $S \to S \times S$

Question 6. Consider the set of natural numbers $\mathbb{N} = \{1, 2, 3, ...\}$. Which of the following is NOT a binary operation on $\mathbb{N}$?

(A) Addition (+)

(B) Multiplication ($\times$)

(C) Division ($\div$)

(D) Maximum (max($a, b$))

Question 7. Which of the following binary operations on $\mathbb{Z}$ is NOT commutative?

(A) $a * b = a + b$

(B) $a * b = a \times b$

(C) $a * b = a - b$

(D) $a * b = \min(a, b)$

Question 8. Which of the following binary operations on $\mathbb{R}$ is NOT associative?

(A) $a * b = a + b$

(B) $a * b = a \times b$

(C) $a * b = a - b$

(D) $a * b = \frac{a+b}{2}$

Question 9. For the binary operation of multiplication ($\times$) on the set of integers $\mathbb{Z}$, which of the following statements is FALSE?

(A) 1 is the identity element.

(B) The inverse of an element $a \neq 0$ is $1/a$.

(C) Every element in $\mathbb{Z}$ has an inverse under multiplication.

(D) The inverse of -1 is -1.

Question 10. Which of the following is NOT a property of a binary operation?

(A) Closure

(B) Uniqueness of identity element (if it exists)

(C) Uniqueness of inverse element (if it exists and the operation is associative and has an identity)

(D) Every binary operation must be commutative and associative.