Single Best Answer MCQs for Sub-Topics of Topic 9: Sets, Relations & Functions

Question 1. Which of the following is a well-defined collection of objects?

(A) The collection of all good cricket players of India.

(B) The collection of all beautiful girls of Delhi.

(C) The collection of all months of a year beginning with the letter 'J'.

(D) The collection of ten most talented writers of the world.

Question 2. Which symbol is used to denote "belongs to" or "is an element of" a set?

(A) $\subset$

(B) $\in$

(C) $\subseteq$

(D) $\notin$

Question 3. The roster form of the set $A = \{x : x \text{ is an even prime number}\}$ is:

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

(B) $\{2\}$

(C) $\{ \}$

(D) $\{0, 2, 4, ...\}$

Question 4. The set-builder form of the set $B = \{1, 4, 9, 16, 25, ...\}$ is:

(A) $\{x : x = n^2, n \in \mathbb{N}\}$

(B) $\{x : x = n^2, n \in \mathbb{Z}\}$

(C) $\{x : x = n^2, n \in \mathbb{W}\}$

(D) $\{x : x = n^2, n \text{ is a positive integer}\}$

Question 5. Let $C = \{x : x^2 - 5x + 6 = 0\}$. The roster form of set C is:

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

(B) $\{-2, -3\}$

(C) $\{1, 5\}$

(D) $\{0, 1\}$

Question 6. The set of natural numbers is denoted by:

(A) $\mathbb{Z}$

(B) $\mathbb{Q}$

(C) $\mathbb{N}$

(D) $\mathbb{R}$

Question 7. Which notation represents the set of all real numbers?

(A) $\mathbb{Z}$

(B) $\mathbb{C}$

(C) $\mathbb{Q}$

(D) $\mathbb{R}$

Question 8. What does the notation $\mathbb{Q}$ represent?

(A) Set of integers

(B) Set of rational numbers

(C) Set of irrational numbers

(D) Set of complex numbers

Question 9. An ordered pair $(a, b)$ is equal to $(c, d)$ if and only if:

(A) $a=c$ and $b=d$

(B) $a=b$ and $c=d$

(C) $a=d$ and $b=c$

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

Question 10. If $(x+1, y-2) = (3, 1)$, then the values of $x$ and $y$ are:

(A) $x=2, y=3$

(B) $x=3, y=2$

(C) $x=4, y=3$

(D) $x=2, y=4$

Question 11. Let $A = \{a, b\}$ and $B = \{1, 2\}$. Find the Cartesian product $A \times B$.

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

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

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

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

Question 12. If $n(A) = 3$ and $n(B) = 2$, what is the number of elements in $A \times B$?

(A) 5

(B) 6

(C) 8

(D) 9

Question 13. If $A = \{1, 2\}$, find $A \times A \times A$.

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

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

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

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

Question 14. Let $A = \{x : x \text{ is a letter in the word 'MATHEMATICS'}\}$. Which of the following elements is NOT in set A?

(A) M

(B) H

(C) P

(D) T

Question 15. The set $\{x \in \mathbb{Z} : -3 < x \leq 1\}$ in roster form is:

(A) $\{-3, -2, -1, 0, 1\}$

(B) $\{-2, -1, 0, 1\}$

(C) $\{-2, -1, 0\}$

(D) $\{-3, -2, -1, 0\}$

Question 16. If $A = \{1, 2\}$ and $B = \{3, 4\}$, then the number of relations from A to B is:

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

(B) $2+2$

(C) $2 \times 2$

(D) $2^{2+2}$

Question 17. The set of all odd integers is represented by:

(A) $\{2k : k \in \mathbb{Z}\}$

(B) $\{2k+1 : k \in \mathbb{Z}\}$

(C) $\{2k-1 : k \in \mathbb{N}\}$

(D) $\{k+1 : k \text{ is an even integer}\}$

Question 18. The set of all prime numbers less than 10 in roster form is:

(A) $\{2, 3, 5, 7, 9\}$

(B) $\{1, 2, 3, 5, 7\}$

(C) $\{2, 3, 5, 7\}$

(D) $\{1, 3, 5, 7, 9\}$

Question 19. If $(x/3 + 1, y - 2/3) = (5/3, 1/3)$, then $(x, y)$ is:

(A) $(4, 1)$

(B) $(1, 4)$

(C) $(-4, -1)$

(D) $(4, -1)$

Question 20. The set of all positive integers is the same as:

(A) $\mathbb{W}$ (Whole numbers)

(B) $\mathbb{Z}^+$ (Positive integers)

(C) $\mathbb{Q}^+$ (Positive rational numbers)

(D) $\mathbb{R}^+$ (Positive real numbers)

Question 1. Which of the following is an empty set?

(A) $\{x : x^2 = 4, x \in \mathbb{Z}\}$

(B) $\{x : x^2 = 2, x \in \mathbb{Q}\}$

(C) $\{x : x^2 = 9, x \in \mathbb{R}\}$

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

Question 2. The set of all points on a line is:

(A) A finite set

(B) An infinite set

(C) An empty set

(D) A singleton set

Question 3. What is the cardinal number of the set $A = \{x : x \text{ is a digit in the number } 150000\}$?

(A) 6

(B) 5

(C) 3

(D) 2

Question 4. Two finite sets A and B are called equivalent if:

(A) $A = B$

(B) $n(A) = n(B)$

(C) $A \subset B$

(D) $B \subset A$

Question 5. If $A = \{1, 2, 3\}$ and $B = \{x : x \text{ is a positive integer and } x^2 < 10\}$, are sets A and B equal?

(A) Yes, because they have the same number of elements.

(B) Yes, because their elements are exactly the same.

(C) No, because B contains 0.

(D) No, because A contains 3 but B contains 1, 2, 3.

Question 6. The number of subsets of a set with $n$ elements is:

(A) $n$

(B) $n^2$

(C) $2^n$

(D) $n!$

Question 7. If $A = \{a, b, c\}$, which of the following is a subset of A?

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

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

(C) $\{ \}$

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

Question 8. If A is a subset of B, then B is called the _______ of A.

(A) Subset

(B) Power set

(C) Universal set

(D) Superset

Question 9. The power set of the empty set $\phi$ is:

(A) $\phi$

(B) $\{ \phi \}$

(C) $\{0\}$

(D) $\{ \{\} \}$

Question 10. If $A = \{1, 2\}$, find $P(A)$, the power set of A.

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

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

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

(D) $\{ \phi, 1, 2, \{1, 2\} \}$

Question 11. In a discussion about students in a classroom, the set of all students in that classroom can be considered as the:

(A) Subset

(B) Power set

(C) Universal set

(D) Empty set

Question 12. The interval $(a, b)$ represents the set of real numbers $x$ such that:

(A) $a \leq x \leq b$

(B) $a < x \leq b$

(C) $a \leq x < b$

(D) $a < x < b$

Question 13. The interval $[a, b]$ represents the set of real numbers $x$ such that:

(A) $a \leq x \leq b$

(B) $a < x \leq b$

(C) $a \leq x < b$

(D) $a < x < b$

Question 14. Which interval notation represents the set $\{x \in \mathbb{R} : x > 5\}$?

(A) $(-\infty, 5)$

(B) $(5, \infty)$

(C) $[5, \infty)$

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

Question 15. What is the cardinal number of the set of integers between 10 and 20 (exclusive)?

(A) 9

(B) 10

(C) 11

(D) 19

Question 16. If A is a finite set with $n$ elements, the number of proper subsets of A is:

(A) $2^n$

(B) $2^n - 1$

(C) $2^n - 2$

(D) $n-1$

Question 17. The set of positive even numbers is:

(A) Finite

(B) Infinite

(C) Empty

(D) Singleton

Question 18. Let $A = \{1, \{2, 3\}\}$. What is the cardinal number of A?

(A) 1

(B) 2

(C) 3

(D) 4

Question 19. If A and B are two sets such that $A \subseteq B$, then $P(A) \subseteq P(B)$ is:

(A) Always true

(B) Always false

(C) True only if A is not empty

(D) True only if B is not the universal set

Question 20. The interval $(- \infty, \infty)$ represents:

(A) The set of natural numbers

(B) The set of rational numbers

(C) The set of real numbers

(D) The empty set

Question 1. If $A = \{1, 2, 3\}$ and $B = \{1, 2, 3, 4\}$, which relation is correct?

(A) $B \subseteq A$

(B) $A \subset B$

(C) $A = B$

(D) $A \supset B$

Question 2. Which symbol represents a proper subset?

(A) $\subset$

(B) $\subseteq$

(C) $\in$

(D) $\supseteq$

Question 3. A relation R from set A to set B is a subset of:

(A) $A \cup B$

(B) $A \cap B$

(C) $A \times B$

(D) $B \times A$

Question 4. If $A = \{1, 2\}$ and $B = \{3, 4\}$, and $R$ is the relation "is less than" from A to B, which ordered pairs are in R?

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

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

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

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

Question 5. Let $A = \{1, 2, 3, 4, 5, 6\}$. A relation R is defined on A as $R = \{(x, y) : y = x + 1\}$. The domain of R is:

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

(B) $\{2, 3, 4, 5, 6, 7\}$

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

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

Question 6. For the relation $R$ in Question 5, the range of R is:

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

(B) $\{2, 3, 4, 5, 6, 7\}$

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

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

Question 7. Let $A = \{1, 2, 3\}$. A relation R on A is given by $R = \{(1, 1), (2, 2), (3, 3)\}$. What is the domain of R?

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

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

(C) $\{3\}$

(D) $\phi$

Question 8. For the relation $R = \{(1, 1), (2, 2), (3, 3)\}$ on $A = \{1, 2, 3\}$, what is the range of R?

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

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

(C) $\{3\}$

(D) $\phi$

Question 9. If a relation R is represented by an arrow diagram from set A to set B, the elements in set A from which arrows originate form the:

(A) Range

(B) Codomain

(C) Domain

(D) Image

Question 10. Let $R$ be a relation from $A=\{a, b\}$ to $B=\{c, d\}$ defined as $R = \{(a, c), (a, d), (b, c)\}$. The range of R is:

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

(B) $\{c, d\}$

(C) $\{a, c, d\}$

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

Question 11. If $A=\{1, 2, 3\}$ and $B=\{p, q\}$, the number of relations from A to B is:

(A) $3^2$

(B) $2^3$

(C) $3 \times 2$

(D) $2^{3 \times 2}$

Question 12. Let R be a relation on $\mathbb{Z}$ defined by $R = \{(x, y) : x - y \text{ is an even integer}\}$. Which of the following pairs belongs to R?

(A) $(3, 5)$

(B) $(4, 7)$

(C) $(2, 3)$

(D) $(1, 0)$

Question 13. Let $A = \{1, 2, 3\}$. The relation $R = \{(1, 2), (2, 3)\}$ on A is a subset of:

(A) A

(B) $A \cup A$

(C) $A \times A$

(D) $A \cap A$

Question 14. The range of a relation is the set of all _________ in the ordered pairs.

(A) First elements

(B) Second elements

(C) Elements from the first set

(D) Elements from the second set

Question 15. Let $A = \{2, 3, 4, 5\}$. Consider the relation R on A defined by $R = \{(x, y) : x \text{ divides } y\}$. The elements of R are:

(A) $\{(2, 2), (3, 3), (4, 4), (5, 5)\}$

(B) $\{(2, 2), (2, 4), (3, 3), (4, 4), (5, 5)\}$

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

(D) $\{(2, 2), (2, 4), (3, 3), (4, 4), (5, 5)\}$

Question 16. For the relation R from Question 15, what is the domain of R?

(A) $\{2, 3, 4, 5\}$

(B) $\{2, 3, 4\}$

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

(D) $\{3, 5\}$

Question 17. For the relation R from Question 15, what is the range of R?

(A) $\{2, 3, 4, 5\}$

(B) $\{2, 3, 4\}$

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

(D) $\{3, 5\}$

Question 18. Let R be a relation "is taller than" from set A (students in Class X) to set B (students in Class IX). If a student 'a' from A is in relation R with a student 'b' from B, this is represented as:

(A) $(a, b) \in R$

(B) $a R b$

(C) Both (A) and (B)

(D) Neither (A) nor (B)

Question 19. If $A=\{x, y\}$ and $B=\{p, q, r\}$, how many ordered pairs are possible in the Cartesian product $A \times B$?

(A) 2

(B) 3

(C) 5

(D) 6

Question 20. Consider the set of integers $\mathbb{Z}$. The relation $R = \{(x, y) : x \leq y\}$ is a relation on $\mathbb{Z}$. Which of the following is NOT true?

(A) $(5, 5) \in R$

(B) $(-2, 0) \in R$

(C) $(3, 1) \in R$

(D) $(-4, -3) \in R$

Question 1. Let $A = \{1, 2, 3\}$. Which of the following is the identity relation on A?

(A) $R_1 = \{(1, 1), (2, 2)\}$

(B) $R_2 = \{(1, 1), (2, 2), (3, 3)\}$

(C) $R_3 = \{(1, 2), (2, 3), (3, 1)\}$

(D) $R_4 = \{(1, 1), (2, 2), (3, 3), (1, 2)\}$

Question 2. A relation R on a set A is called reflexive if for every element $a \in A$, _______.

(A) $(a, a) \in R$

(B) $(a, a) \notin R$

(C) $(a, b) \in R$ for some $b \in A$

(D) $(b, a) \in R$ for some $b \in A$

Question 3. Let A be the set of all lines in a plane. The relation $R$ on A defined by $R = \{(l_1, l_2) : l_1 \text{ is parallel to } l_2\}$ is:

(A) Reflexive

(B) Symmetric

(C) Transitive

(D) All of the above

Question 4. A relation R on a set A is symmetric if for every $(a, b) \in R$, _______.

(A) $(a, a) \in R$

(B) $(b, b) \in R$

(C) $(b, a) \in R$

(D) $(a, c) \in R$ for some $c$

Question 5. Let A be the set of all lines in a plane. The relation $R$ on A defined by $R = \{(l_1, l_2) : l_1 \text{ is perpendicular to } l_2\}$ is:

(A) Reflexive

(B) Symmetric

(C) Transitive

(D) Reflexive and Transitive

Question 6. A relation R on a set A is transitive if for every $(a, b) \in R$ and $(b, c) \in R$, it implies _______.

(A) $(a, a) \in R$

(B) $(b, b) \in R$

(C) $(c, a) \in R$

(D) $(a, c) \in R$

Question 7. A relation which is reflexive, symmetric, and transitive is called a(n):

(A) Identity relation

(B) Equivalence relation

(C) Universal relation

(D) Empty relation

Question 8. Let $A = \{1, 2, 3\}$. Consider the relation $R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)\}$. Is R reflexive?

(A) Yes

(B) No

(C) Cannot be determined

(D) Only if A was an infinite set

Question 9. For the relation R in Question 8, is R symmetric?

(A) Yes

(B) No

(C) Cannot be determined

(D) Only if A was an infinite set

Question 10. For the relation R in Question 8, is R transitive?

(A) Yes

(B) No

(C) Cannot be determined

(D) Only if A was an infinite set

Question 11. Let R be the relation "is brother of" on the set of all people. This relation is:

(A) Reflexive, Symmetric, Transitive

(B) Symmetric

(C) Transitive

(D) None of the above (assuming strict biological relation and no person is a brother of themselves)

Question 12. The empty relation on a non-empty set A is:

(A) Always reflexive

(B) Never reflexive

(C) Sometimes reflexive

(D) Reflexive only if A is a singleton set

Question 13. The universal relation on a non-empty set A is:

(A) Always reflexive

(B) Never reflexive

(C) Sometimes reflexive

(D) Reflexive only if A is an infinite set

Question 14. Let $A = \{1, 2, 3\}$. The relation $R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 3)\}$. Is R an equivalence relation?

(A) Yes

(B) No

(C) Cannot be determined

(D) Yes, if A was an infinite set

Question 15. Let R be the relation "is congruent to" on the set of all triangles in a plane. This relation is:

(A) Reflexive only

(B) Symmetric only

(C) Transitive only

(D) An equivalence relation

Question 16. Let R be an equivalence relation on a set A. The set of all elements related to a specific element $a \in A$ is called the _______ of $a$.

(A) Domain

(B) Range

(C) Equivalence class

(D) Subset

Question 17. Consider the relation R on $\mathbb{Z}$ defined by $a R b$ if $a-b$ is divisible by 5. This relation is:

(A) Symmetric but not reflexive

(B) Transitive but not symmetric

(C) Reflexive, symmetric, and transitive

(D) Neither reflexive, symmetric, nor transitive

Question 18. For the equivalence relation in Question 17, the equivalence class of 0 is:

(A) $\{0, 5, 10, ...\}$

(B) $\{..., -5, 0, 5, 10, ...\}$

(C) $\{0\}$

(D) $\mathbb{Z}$

Question 19. Let $A = \{a, b, c\}$. Which of the following relations on A is symmetric but not reflexive?

(A) $R_1 = \{(a, a), (b, b)\}$

(B) $R_2 = \{(a, b), (b, a)\}$

(C) $R_3 = \{(a, a), (b, b), (c, c)\}$

(D) $R_4 = \{(a, b), (b, a), (c, c)\}$

Question 20. Let R be the relation "is less than or equal to" on the set of real numbers $\mathbb{R}$. This relation is:

(A) Reflexive and Symmetric

(B) Symmetric and Transitive

(C) Reflexive and Transitive

(D) An equivalence relation

Question 1. Let $A = \{1, 2, 3, 4\}$ and $B = \{3, 4, 5, 6\}$. Find $A \cup B$.

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

(B) $\{3, 4\}$

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

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

Question 2. Let $A = \{1, 2, 3, 4\}$ and $B = \{3, 4, 5, 6\}$. Find $A \cap B$.

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

(B) $\{3, 4\}$

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

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

Question 3. Let $A = \{1, 2, 3, 4\}$ and $B = \{3, 4, 5, 6\}$. Find $A - B$.

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

(B) $\{5, 6\}$

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

(D) $\{3, 4\}$

Question 4. Let $A = \{1, 2, 3, 4\}$ and $B = \{3, 4, 5, 6\}$. Find $B - A$.

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

(B) $\{5, 6\}$

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

(D) $\{3, 4\}$

Question 5. Let U be the universal set and A be any set. The complement of A, denoted by $A'$, is given by:

(A) $A \cup U$

(B) $A \cap U$

(C) $U - A$

(D) $A - U$

Question 6. If U is the set of all integers and $A$ is the set of all even integers, then $A'$ (complement of A) is the set of:

(A) Odd integers

(B) Prime numbers

(C) All integers

(D) Empty set

Question 7. In a Venn diagram, the universal set is usually represented by a:

(A) Circle

(B) Rectangle

(C) Triangle

(D) Line

Question 8. The shaded region in a Venn diagram representing $A \cap B$ corresponds to the elements that are:

(A) In A or B or both

(B) In A and B

(C) In A but not in B

(D) Not in A and not in B

Question 9. The shaded region in a Venn diagram representing $A \cup B$ corresponds to the elements that are:

(A) In A or B or both

(B) In A and B

(C) In A but not in B

(D) Not in A and not in B

Question 10. Which set operation is represented by the region outside a circle representing set A, within the universal rectangle?

(A) $A \cup A'$

(B) $A \cap A'$

(C) $A'$

(D) U

Question 11. If A and B are disjoint sets, then $A \cap B$ is:

(A) $A$

(B) $B$

(C) $U$

(D) $\phi$

Question 12. For any set A, $A \cup \phi$ is equal to:

(A) $\phi$

(B) A

(C) U

(D) $A'$

Question 13. For any set A, $A \cap U$ is equal to:

(A) $\phi$

(B) A

(C) U

(D) $A'$

Question 14. If $A \subseteq B$, then $A \cap B$ is equal to:

(A) A

(B) B

(C) $\phi$

(D) U

Question 15. If $A \subseteq B$, then $A \cup B$ is equal to:

(A) A

(B) B

(C) $\phi$

(D) U

Question 16. $(A')'$ is equal to:

(A) A

(B) $A'$

(C) U

(D) $\phi$

Question 17. $U'$ is equal to:

(A) U

(B) $\phi$

(C) $U'$

(D) Any set A

Question 18. $\phi'$ is equal to:

(A) U

(B) $\phi$

(C) $U'$

(D) Any set A

Question 19. The shaded region in the Venn diagram below represents:

(A) $A \cup B$

(B) $A \cap B$

(C) $A - B$

(D) $B - A$

Question 20. The shaded region in the Venn diagram below represents:

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

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

(C) $A' \cup B'$

(D) $A - B'$

Question 1. For any three sets A, B, and C, $A \cup (B \cap C)$ is equal to:

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

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

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

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

Question 2. This property $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ is known as the:

(A) Commutative Law

(B) Associative Law

(C) Distributive Law

(D) Idempotent Law

Question 3. De Morgan's Law states that $(A \cup B)'$ is equal to:

(A) $A' \cup B'$

(B) $A' \cap B'$

(C) $A \cup B$

(D) $A \cap B$

Question 4. De Morgan's Law also states that $(A \cap B)'$ is equal to:

(A) $A' \cup B'$

(B) $A' \cap B'$

(C) $A \cup B$

(D) $A \cap B$

Question 5. For two finite sets A and B, $n(A \cup B)$ is equal to:

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

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

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

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

Question 6. If A and B are disjoint finite sets, then $n(A \cup B)$ is equal to:

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

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

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

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

Question 7. In a group of 70 people, 37 like coffee, 52 like tea, and each person likes at least one of the two drinks. How many people like both coffee and tea?

(A) 15

(B) 19

(C) 35

(D) 52

Question 8. In a survey of 100 students, the number of students studying different subjects are as follows: Mathematics 28, Physics 30, Chemistry 22, Mathematics and Physics 8, Mathematics and Chemistry 10, Physics and Chemistry 5, All three subjects 3. How many students study Mathematics only?

(A) 28

(B) 13

(C) 20

(D) 15

Question 9. For three finite sets A, B, and C, $n(A \cup B \cup C)$ is given by:

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

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

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

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

Question 10. $A \cap A'$ is equal to:

(A) A

(B) $A'$

(C) U

(D) $\phi$

Question 11. $A \cup A'$ is equal to:

(A) A

(B) $A'$

(C) U

(D) $\phi$

Question 12. Which law is represented by $A \cup \phi = A$?

(A) Identity Law

(B) Commutative Law

(C) Associative Law

(D) Idempotent Law

Question 13. Which law is represented by $A \cap A = A$?

(A) Identity Law

(B) Commutative Law

(C) Associative Law

(D) Idempotent Law

Question 14. If $n(A) = 20$, $n(B) = 30$, and $n(A \cap B) = 10$, find $n(A \cup B)$.

(A) 40

(B) 50

(C) 60

(D) 10

Question 15. In a committee, 50 people speak Hindi, 20 speak English, and 10 speak both Hindi and English. How many people speak at least one of these two languages?

(A) 60

(B) 70

(C) 80

(D) 40

Question 16. If $n(U) = 100$, $n(A) = 60$, $n(B) = 50$, and $n(A \cap B) = 20$, find $n((A \cup B)')$.

(A) 70

(B) 30

(C) 40

(D) 80

Question 17. The principle of inclusion-exclusion for two sets is primarily used to find the cardinality of the:

(A) Union of the sets

(B) Intersection of the sets

(C) Difference of the sets

(D) Complement of a set

Question 18. If $A$ and $B$ are two sets, then $n(A - B)$ is equal to:

(A) $n(A) - n(B)$

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

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

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

Question 19. In a class of 35 students, 24 like to play cricket and 16 like to play football. Also, each student likes to play at least one of the two games. How many students like to play only cricket?

(A) 24

(B) 16

(C) 8

(D) 19

Question 20. Given $n(A \cup B) = 50$, $n(A) = 28$, $n(B) = 32$. Find $n(A \cap B)$.

(A) 10

(B) 50

(C) 60

(D) 0

Question 1. Which of the following relations from $A=\{1, 2, 3\}$ to $B=\{a, b, c\}$ is a function?

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

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

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

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

Question 2. A relation $f$ from a set A to a set B is called a function if:

(A) Every element of A is related to at least one element of B.

(B) Every element of A is related to exactly one element of B.

(C) Every element of B is related to some element of A.

(D) Every element of B is related to exactly one element of A.

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

(A) Codomain

(B) Range

(C) Domain

(D) Image

Question 4. If $f: A \to B$ is a function, then set B is called the:

(A) Codomain

(B) Range

(C) Domain

(D) Pre-image

Question 5. The range of a function $f: A \to B$ is the set of all:

(A) Elements in A

(B) Elements in B

(C) Images of elements of A under f

(D) Elements in B that have a pre-image in A

Question 6. Let $f = \{(1, 2), (2, 3), (3, 4), (4, 5)\}$ be a function from $A=\{1, 2, 3, 4\}$ to $B=\{1, 2, 3, 4, 5\}$. The domain of $f$ is:

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

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

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

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

Question 7. For the function $f$ in Question 6, the range of $f$ is:

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

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

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

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

Question 8. If a function is defined by $f(x) = 2x + 1$, and its domain is $\{1, 2, 3\}$, what is the range of the function?

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

(B) $\{3, 5, 7\}$

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

(D) $\{1, 3, 5, 7\}$

Question 9. Which of the following describes a function rule $f(x) = |x|$?

(A) Assigns each real number to itself.

(B) Assigns each positive number to itself and each negative number to its positive counterpart.

(C) Assigns each number to its square.

(D) Assigns each number to 0.

Question 10. The key difference between a relation and a function from set A to set B is that in a function:

(A) Every element in A must be mapped to some element in B.

(B) Every element in the domain has a unique image.

(C) An element in A can be mapped to more than one element in B.

(D) There must be at least one element in A that is not mapped.

Question 11. If $f(x) = x^2 + 1$, and the domain is $\{-1, 0, 1, 2\}$, the range of $f$ is:

(A) $\{0, 1, 2, 5\}$

(B) $\{1, 2, 5\}$

(C) $\{-1, 0, 1, 2\}$

(D) $\{1, 0, 1, 5\}$

Question 12. If a function $f: A \to B$ is such that its range is equal to its codomain, the function is called:

(A) One-to-one

(B) Onto

(C) Bijective

(D) Identity

Question 13. The set of all possible outputs of a function is called its:

(A) Domain

(B) Codomain

(C) Range

(D) Graph

Question 14. If $f: \mathbb{N} \to \mathbb{N}$ is defined by $f(n) = n+1$, the domain of $f$ is:

(A) $\mathbb{W}$

(B) $\mathbb{Z}$

(C) $\mathbb{N}$

(D) $\mathbb{R}$

Question 15. For the function $f(n) = n+1$ with domain $\mathbb{N}$, the range of $f$ is:

(A) $\mathbb{N}$

(B) $\{1, 2, 3, ...\}$

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

(D) $\mathbb{W}$

Question 16. If the range of a function $f: A \to B$ is a proper subset of its codomain B, the function is called:

(A) Onto

(B) Into

(C) Bijective

(D) Identity

Question 17. Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = x^2$. The domain of $f$ is:

(A) $\mathbb{N}$

(B) $\mathbb{Z}$

(C) $\mathbb{Q}$

(D) $\mathbb{R}$

Question 18. For the function $f(x) = x^2$ with domain $\mathbb{R}$ and codomain $\mathbb{R}$, the range of $f$ is:

(A) $\mathbb{R}$

(B) $[0, \infty)$

(C) $(-\infty, 0]$

(D) $(0, \infty)$

Question 19. If $f: A \to B$ is a function, and $(a, b) \in f$, then 'b' is called the _______ of 'a' under f.

(A) Pre-image

(B) Domain element

(C) Image

(D) Codomain element

Question 20. Which of the following properties is essential for a relation to be a function?

(A) Every element in the codomain has a pre-image.

(B) Every element in the domain has a unique image.

(C) The relation is reflexive.

(D) The relation is symmetric.

Question 1. A function $f: A \to B$ is called one-to-one (injective) if:

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

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

(C) Every element in A has a unique image and every element in B has a unique pre-image.

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

Question 2. A function $f: A \to B$ is called onto (surjective) if:

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

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

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

(D) The range of $f$ is a subset of B.

Question 3. A function is bijective if it is both:

(A) One-to-one and Into

(B) Many-to-one and Onto

(C) One-to-one and Onto

(D) Many-to-one and Into

Question 4. Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = 2x + 3$. This function is:

(A) One-to-one but not onto

(B) Onto but not one-to-one

(C) Both one-to-one and onto (bijective)

(D) Neither one-to-one nor onto

Question 5. Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = x^2$. This function is:

(A) One-to-one but not onto

(B) Onto but not one-to-one

(C) Both one-to-one and onto

(D) Neither one-to-one nor onto

Question 6. A function $f: A \to B$ is called many-to-one if:

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

(B) At least two different elements in A have the same image in B.

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

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

Question 7. A function $f: A \to B$ is called into if:

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

(B) The range of $f$ is a proper subset of the codomain B.

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

(D) Every element in A has a unique image.

Question 8. Let $f: \mathbb{N} \to \mathbb{N}$ be defined by $f(n) = n^2$. This function is:

(A) One-to-one and onto

(B) One-to-one and into

(C) Many-to-one and onto

(D) Many-to-one and into

Question 9. Let $f: \mathbb{R} \to [-1, 1]$ be defined by $f(x) = \sin(x)$. This function is:

(A) One-to-one and onto

(B) One-to-one and into

(C) Many-to-one and onto

(D) Many-to-one and into

Question 10. If $f: A \to B$ is a function where $n(A) = 3$ and $n(B) = 2$. Which of the following types of functions is NOT possible for $f$?

(A) Many-to-one

(B) Onto

(C) One-to-one

(D) Into

Question 11. If $f: A \to B$ is a function where $n(A) = 2$ and $n(B) = 3$. Which of the following types of functions is NOT possible for $f$?

(A) One-to-one

(B) Into

(C) Onto

(D) Both A and B are possible

Question 12. A function $f: A \to B$ is one-to-one if for all $x_1, x_2 \in A$, $f(x_1) = f(x_2)$ implies:

(A) $x_1 \neq x_2$

(B) $x_1 = x_2$

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

(D) $x_1 = -x_2$

Question 13. Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = |x|$. This function is:

(A) One-to-one and onto

(B) One-to-one and into

(C) Many-to-one and onto

(D) Many-to-one and into

Question 14. Which of the following is a criterion for a function to be onto?

(A) The domain equals the codomain.

(B) The range equals the codomain.

(C) Each element in the domain has a unique image.

(D) The number of elements in the domain is less than or equal to the number of elements in the codomain.

Question 15. Let $f: \mathbb{Z} \to \mathbb{Z}$ be defined by $f(x) = x+1$. This function is:

(A) One-to-one and onto

(B) One-to-one and into

(C) Many-to-one and onto

(D) Many-to-one and into

Question 16. Let $A = \{1, 2, 3\}$ and $B = \{p, q\}$. A function $f: A \to B$ can be:

(A) One-to-one

(B) Bijective

(C) Onto

(D) Into

Question 17. If $f: A \to B$ is a surjective function, it means:

(A) Every element of A is mapped to some element of B.

(B) The number of elements in A is less than or equal to the number of elements in B.

(C) The number of elements in A is greater than or equal to the number of elements in B.

(D) The range of f is A.

Question 18. Consider the function $f = \{(1, a), (2, b), (3, a)\}$ from $A=\{1, 2, 3\}$ to $B=\{a, b, c\}$. This function is:

(A) One-to-one and onto

(B) One-to-one and into

(C) Many-to-one and onto

(D) Many-to-one and into

Question 19. A function that is both injective and surjective is called:

(A) Endomorphism

(B) Isomorphism

(C) Automorphism

(D) Homomorphism

Question 20. If $f: A \to B$ is a function, and $n(A) = n(B)$ is finite, then $f$ is one-to-one if and only if $f$ is:

(A) Into

(B) Many-to-one

(C) Onto

(D) Constant

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

(A) $\mathbb{N}$

(B) $\mathbb{Z}$

(C) $\mathbb{Q}$

(D) $\mathbb{R}$

Question 2. The domain of the real function $f(x) = \sqrt{x-1}$ is:

(A) $x \geq 1$

(B) $x > 1$

(C) $x \leq 1$

(D) $x < 1$

Question 3. The range of the real function $f(x) = \sqrt{x-1}$ is:

(A) $(-\infty, \infty)$

(B) $[0, \infty)$

(C) $(1, \infty)$

(D) $[1, \infty)$

Question 4. The domain of the real function $f(x) = \frac{1}{x+2}$ is:

(A) $\mathbb{R}$

(B) $\mathbb{R} - \{0\}$

(C) $\mathbb{R} - \{-2\}$

(D) $\mathbb{R} - \{2\}$

Question 5. The range of the real function $f(x) = \frac{1}{x+2}$ is:

(A) $\mathbb{R}$

(B) $\mathbb{R} - \{0\}$

(C) $\mathbb{R} - \{-2\}$

(D) $\mathbb{R} - \{2\}$

Question 6. The graph of a function $y = f(x)$ is a set of all points $(x, y)$ such that:

(A) $x$ is in the codomain and $y$ is in the domain.

(B) $x$ is in the domain and $y$ is in the range, and $y = f(x)$.

(C) $x$ is in the domain and $y$ is in the codomain.

(D) $x$ is in the range and $y$ is in the domain.

Question 7. The vertical line test is used to determine if a given graph represents a:

(A) Relation

(B) Function

(C) One-to-one function

(D) Onto function

Question 8. The horizontal line test is used to determine if a given graph represents a:

(A) Relation

(B) Function

(C) One-to-one function

(D) Onto function

Question 9. The graph of the identity function $f(x) = x$ is a straight line passing through the origin with slope:

(A) 0

(B) 1

(C) -1

(D) Undefined

Question 10. The range of the modulus function $f(x) = |x|$ is:

(A) $\mathbb{R}$

(B) $[0, \infty)$

(C) $(-\infty, 0]$

(D) $\{0\}$

Question 11. The graph of the function $f(x) = x^2$ is a:

(A) Straight line

(B) Circle

(C) Parabola

(D) Hyperbola

Question 12. The domain of the function $f(x) = \frac{1}{\sqrt{x-5}}$ is:

(A) $x \geq 5$

(B) $x > 5$

(C) $x \leq 5$

(D) $x < 5$

Question 13. The graph of the constant function $f(x) = c$ is a:

(A) Vertical line

(B) Horizontal line

(C) Line passing through the origin

(D) Parabola

Question 14. The domain of the function $f(x) = |x-3|$ is:

(A) $[3, \infty)$

(B) $(-\infty, 3]$

(C) $\mathbb{R}$

(D) $\mathbb{R} - \{3\}$

Question 15. The range of the function $f(x) = |x-3|$ is:

(A) $[3, \infty)$

(B) $(-\infty, 3]$

(C) $\mathbb{R}$

(D) $[0, \infty)$

Question 16. The graph of the function $f(x) = x^3$ is symmetric about the:

(A) X-axis

(B) Y-axis

(C) Origin

(D) Line $y=x$

Question 17. The domain of the function $f(x) = \sin(x)$ is:

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

(B) $[-1, 1]$

(C) $\mathbb{R}$

(D) $(-\infty, \infty) - \{n\pi\}$

Question 18. The range of the function $f(x) = \cos(x)$ is:

(A) $\mathbb{R}$

(B) $[-1, 1]$

(C) $(0, \infty)$

(D) $(-\infty, \infty) - \{ (2n+1)\frac{\pi}{2} \}$

Question 19. For the function $f(x) = \begin{cases} 2x & , & x < 0 \\ x^2 & , & x \geq 0 \end{cases}$, what is $f(-3)$?

(A) 9

(B) -9

(C) 6

(D) -6

Question 20. For the function $f(x) = \begin{cases} 2x & , & x < 0 \\ x^2 & , & x \geq 0 \end{cases}$, what is $f(3)$?

(A) 9

(B) -9

(C) 6

(D) -6

Question 1. Let $f(x) = x^2$ and $g(x) = x + 1$. Find $(f+g)(x)$.

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

(B) $x^2 + x$

(C) $2x^2 + 1$

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

Question 2. Let $f(x) = x^2$ and $g(x) = x + 1$. Find $(f-g)(x)$.

(A) $x^2 - x - 1$

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

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

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

Question 3. Let $f(x) = x^2$ and $g(x) = x + 1$. Find $(fg)(x)$.

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

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

(C) $x^3 + 1$

(D) $x^2 + x$

Question 4. Let $f(x) = x^2$ and $g(x) = x + 1$. Find $(f/g)(x)$.

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

(B) $\frac{x+1}{x^2}$

(C) $\frac{x^2}{x+1}$

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

Question 5. The domain of $(f+g)(x)$ is the intersection of the domains of f and g, provided the intersection is:

(A) Empty

(B) Finite

(C) Non-empty

(D) Infinite

Question 6. The domain of $(f/g)(x)$ is the intersection of the domains of f and g, excluding the values of x where:

(A) $f(x) = 0$

(B) $g(x) = 0$

(C) $f(x) + g(x) = 0$

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

Question 7. Let $f(x) = 2x$ and $g(x) = x^2$. Find $(f \circ g)(x)$.

(A) $(2x)^2$

(B) $2(x^2)$

(C) $2x + x^2$

(D) $x^2 + 2x$

Question 8. Let $f(x) = 2x$ and $g(x) = x^2$. Find $(g \circ f)(x)$.

(A) $(2x)^2$

(B) $2(x^2)$

(C) $2x + x^2$

(D) $x^2 + 2x$

Question 9. If $f(x) = x^2$ and $g(x) = \sin x$, then $(f \circ g)(x)$ is:

(A) $\sin(x^2)$

(B) $(\sin x)^2$

(C) $x^2 \sin x$

(D) $\sin x^2$

Question 10. If $f(x) = x^2$ and $g(x) = \sin x$, then $(g \circ f)(x)$ is:

(A) $\sin(x^2)$

(B) $(\sin x)^2$

(C) $x^2 \sin x$

(D) $\sin^2 x$

Question 11. The composition of functions $f \circ g$ is defined if the range of g is a subset of the _______ of f.

(A) Range

(B) Codomain

(C) Domain

(D) Graph

Question 12. Let $f(x) = x+1$ and $g(x) = x-1$. Find $(f \circ g)(x)$.

(A) x

(B) $x+2$

(C) $x-2$

(D) $x^2-1$

Question 13. Let $f(x) = x+1$ and $g(x) = x-1$. Find $(g \circ f)(x)$.

(A) x

(B) $x+2$

(C) $x-2$

(D) $x^2-1$

Question 14. If $(f \circ g)(x) = x$ for all $x$ in the domain of $g$, and $(g \circ f)(x) = x$ for all $x$ in the domain of $f$, then f and g are ________ of each other.

(A) Composites

(B) Sums

(C) Differences

(D) Inverses

Question 15. The operation of composition of functions is generally:

(A) Commutative

(B) Associative

(C) Both commutative and associative

(D) Neither commutative nor associative

Question 16. Let $f(x) = 3x$ and $g(x) = x/3$. Find $(f \circ g)(x)$.

(A) 1

(B) 9x

(C) x

(D) x/9

Question 17. Let $f(x) = 3x$ and $g(x) = x/3$. Find $(g \circ f)(x)$.

(A) 1

(B) 9x

(C) x

(D) x/9

Question 18. If $f(x) = x+2$ and $g(x) = x^2$, what is the domain of $(f/g)(x)$?

(A) $\mathbb{R}$

(B) $\mathbb{R} - \{0\}$

(C) $\mathbb{R} - \{-2\}$

(D) $\mathbb{R} - \{0, -2\}$

Question 19. Let $f(x) = \sqrt{x}$ and $g(x) = x^2$. Find the domain of $(f \circ g)(x) = f(g(x)) = f(x^2) = \sqrt{x^2} = |x|$.

(A) $[0, \infty)$

(B) $\mathbb{R}$

(C) $(-\infty, 0]$

(D) $[0, \infty)$ excluding 0

Question 20. Let $f(x) = x^2$ and $g(x) = \sqrt{x}$. Find the domain of $(f \circ g)(x) = f(g(x)) = f(\sqrt{x}) = (\sqrt{x})^2 = x$.

(A) $\mathbb{R}$

(B) $[0, \infty)$

(C) $(-\infty, 0]$

(D) $(0, \infty)$

Question 1. A function $f: X \to Y$ is said to be invertible if there exists a function $g: Y \to X$ such that:

(A) $f \circ g = I_Y$ and $g \circ f = I_X$

(B) $f \circ g = I_X$ and $g \circ f = I_Y$

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

(D) $f(x) = g(x)$ for all 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 $f$ is invertible, its inverse function $f^{-1}$ is:

(A) Unique

(B) Not unique

(C) Always the identity function

(D) Always the zero function

Question 4. Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = 3x - 5$. To find the inverse function $f^{-1}(y)$, we set $y = f(x)$ and solve for x in terms of y. What is the first step?

(A) $y = 3x - 5$

(B) $y+5 = 3x$

(C) $x = (y+5)/3$

(D) $f^{-1}(y) = (y+5)/3$

Question 5. For the function $f(x) = 3x - 5$, the inverse function $f^{-1}(x)$ is:

(A) $\frac{x+5}{3}$

(B) $\frac{x-5}{3}$

(C) $5x - 3$

(D) $3x + 5$

Question 6. Which of the following functions from $\mathbb{R} \to \mathbb{R}$ is invertible?

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

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

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

(D) $f(x) = 2x + 7$

Question 7. A binary operation $*$ on a set S is a function from:

(A) $S \to S$

(B) $S \times S \to S$

(C) $S \to S \times S$

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

Question 8. A set S is closed under a binary operation $*$ if for all $a, b \in S$, $a * b$ is also in:

(A) The power set of S

(B) The universal set

(C) S

(D) The empty set

Question 9. A binary operation $*$ on a set S is commutative if for all $a, b \in S$, _______.

(A) $a * a = a$

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

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

(D) $a * e = a$ for some element $e$

Question 10. Addition (+) on the set of integers $\mathbb{Z}$ is:

(A) Commutative but not associative

(B) Associative but not commutative

(C) Both commutative and associative

(D) Neither commutative nor associative

Question 11. A binary operation $*$ on a set S is associative if for all $a, b, c \in S$, _______.

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

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

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

(D) $a * e = a$ for some element $e$

Question 12. An element $e \in S$ is called an identity element for a binary operation $*$ on S if for all $a \in S$, _______.

(A) $a * a = e$

(B) $a * e = a$ and $e * a = a$

(C) $a * e = e$ and $e * a = e$

(D) $a * e = a$ only

Question 13. For the binary operation of addition (+) on integers $\mathbb{Z}$, the identity element is:

(A) 0

(B) 1

(C) -1

(D) No identity element exists

Question 14. For a binary operation $*$ on S with identity element $e$, an element $b \in S$ is called the inverse of $a \in S$ if _______.

(A) $a * b = e$

(B) $a * b = b * a = e$

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

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

Question 15. For the binary operation of addition (+) on integers $\mathbb{Z}$, the inverse of an element $a$ is:

(A) $1/a$

(B) $-a$

(C) $a$

(D) 0

Question 16. Let $*$ be a binary operation on $\mathbb{R}$ defined by $a * b = a + 2b$. Is $*$ commutative?

(A) Yes

(B) No

(C) Only for $b=0$

(D) Only for $a=b$

Question 17. Let $*$ be a binary operation on $\mathbb{Q}$ defined by $a * b = \frac{a+b}{2}$. Is $*$ associative?

(A) Yes

(B) No

(C) Only for $a=b=c$

(D) Cannot be determined

Question 18. Which of the following functions from $\mathbb{Z} \to \mathbb{Z}$ is not invertible?

(A) $f(x) = x + 5$

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

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

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

Question 19. If $f: A \to B$ and $g: B \to A$ are two functions such that $g \circ f = I_A$, then f is:

(A) Surjective

(B) Injective

(C) Bijective

(D) Neither injective nor surjective

Question 20. If $f: A \to B$ and $g: B \to A$ are two functions such that $f \circ g = I_B$, then f is:

(A) Surjective

(B) Injective

(C) Bijective

(D) Neither injective nor surjective