1. What is the Cartesian product of two sets A and B?
दो समुच्चयों A और B का Cartesian Product क्या होता है?

A. A × B = A ∪ B / A × B = A ∪ B
B. A × B = {(a,b) : a ∈ A, b ∈ B} / A × B = {(a,b) : a ∈ A, b ∈ B}
C. A × B = A ∩ B / A × B = A ∩ B
D. A × B = A − B / A × B = A − B

Answer: B. {(a,b) : a ∈ A, b ∈ B} / {(a,b) : a ∈ A, b ∈ B}

2. If |A| = 3 and |B| = 4, then |A × B| is:
यदि |A| = 3 और |B| = 4 है, तो |A × B| कितना होगा?

A. 7 / 7
B. 9 / 9
C. 10 / 10
D. 12 / 12

Answer: D. 12 / 12

3. Which of the following conditions must a relation satisfy to be an equivalence relation?
निम्नलिखित में से किन गुणों को किसी संबंध में equivalence relation होने के लिए होना आवश्यक है?

A. Reflexive, symmetric and transitive / Reflexive, symmetric और transitive
B. Reflexive, antisymmetric and transitive / Reflexive, antisymmetric और transitive
C. Symmetric and antisymmetric only / केवल symmetric और antisymmetric
D. Transitive only / केवल transitive

Answer: A. Reflexive, symmetric and transitive / Reflexive, symmetric और transitive

4. A partition of a set consists of:
किसी समुच्चय का partition किससे मिलकर बनता है?

A. Overlapping subsets only / केवल overlapping subsets
B. Empty subsets only / केवल empty subsets
C. Non-empty pairwise disjoint subsets whose union is the set / ऐसे non-empty pairwise disjoint subsets जिनका union पूरा set हो
D. One subset only / केवल एक subset

Answer: C. Non-empty pairwise disjoint subsets whose union is the set / ऐसे non-empty pairwise disjoint subsets जिनका union पूरा set हो

5. Which of the following properties define a partial order relation?
निम्नलिखित में से कौन-से गुण partial order relation को परिभाषित करते हैं?

A. Reflexive, antisymmetric and transitive / Reflexive, antisymmetric और transitive
B. Reflexive, symmetric and transitive / Reflexive, symmetric और transitive
C. Symmetric and transitive only / केवल symmetric और transitive
D. Antisymmetric and symmetric only / केवल antisymmetric और symmetric

Answer: A. Reflexive, antisymmetric and transitive / Reflexive, antisymmetric और transitive

6. In a totally ordered set, any two elements are:
Totally ordered set में किन्हीं भी दो elements के बीच क्या होता है?

A. Always equal / हमेशा equal
B. Always unequal / हमेशा unequal
C. Never comparable / कभी comparable नहीं
D. Comparable / Comparable

Answer: D. Comparable / Comparable

7. A relation f from A to B is a function if:
A से B तक कोई relation f function कहलाता है, यदि:

A. Every element of B has many images / B के प्रत्येक element की कई images हों
B. No element of A has an image / A के किसी element की image न हो
C. Every element of A has exactly one image in B / A के प्रत्येक element की B में exactly one image हो
D. Every element of B has exactly one pre-image / B के प्रत्येक element की exactly one pre-image हो

Answer: C. Every element of A has exactly one image in B / A के प्रत्येक element की B में exactly one image हो

8. If f: A → B and g: B → C, then the composition g ∘ f is defined by:
यदि f: A → B और g: B → C है, तो composition g ∘ f किस प्रकार परिभाषित होता है?

A. (g ∘ f)(x) = f(g(x)) / (g ∘ f)(x) = f(g(x))
B. (g ∘ f)(x) = g(f(x)) / (g ∘ f)(x) = g(f(x))
C. (g ∘ f)(x) = g(x) + f(x) / (g ∘ f)(x) = g(x) + f(x)
D. (g ∘ f)(x) = g(x) − f(x) / (g ∘ f)(x) = g(x) − f(x)

Answer: B. (g ∘ f)(x) = g(f(x)) / (g ∘ f)(x) = g(f(x))

9. A function f: A → B is injective if:
कोई function f: A → B injective कहलाता है, यदि:

A. Different elements of A have different images in B / A के अलग-अलग elements की B में अलग-अलग images हों
B. Every element of B has an image in A / B के प्रत्येक element की A में image हो
C. All elements of A have the same image / A के सभी elements की same image हो
D. A and B are equal sets / A और B equal sets हों

Answer: A. Different elements of A have different images in B / A के अलग-अलग elements की B में अलग-अलग images हों

10. A function f: A → B is surjective if:
कोई function f: A → B surjective कहलाता है, यदि:

A. Every element of A has the same image / A के प्रत्येक element की same image हो
B. A has no elements / A में कोई element न हो
C. Every element of B has at least one pre-image in A / B के प्रत्येक element की A में कम-से-कम एक pre-image हो
D. Every element of A has two images / A के प्रत्येक element की दो images हों

Answer: C. Every element of B has at least one pre-image in A / B के प्रत्येक element की A में कम-से-कम एक pre-image हो

11. A function which is both injective and surjective is called:
जो function injective और surjective दोनों होता है, उसे क्या कहते हैं?

A. Constant function / Constant function
B. Bijective function / Bijective function
C. Identity relation / Identity relation
D. Partial function / Partial function

Answer: B. Bijective function / Bijective function

12. A function has an inverse function if and only if it is:
किसी function का inverse function तभी और केवल तभी होता है जब वह:

A. Bijective / Bijective
B. Constant / Constant
C. Only surjective / केवल surjective
D. Only injective / केवल injective

Answer: A. Bijective / Bijective

13. Which of the following is a countable set?
निम्नलिखित में से कौन-सा countable set है?

A. Set of all real numbers / सभी real numbers का set
B. Set of all irrational numbers / सभी irrational numbers का set
C. Set of integers ℤ / Integers ℤ का set
D. Set of all real numbers between 0 and 1 / 0 और 1 के बीच सभी real numbers का set

Answer: C. Set of integers ℤ / Integers ℤ का set

14. Which of the following statements about the set of rational numbers ℚ is correct?
Rational numbers ℚ के set के बारे में निम्नलिखित में से कौन-सा कथन सही है?

A. ℚ is finite / ℚ finite है
B. ℚ is countable / ℚ countable है
C. ℚ is uncountable / ℚ uncountable है
D. ℚ has no elements / ℚ में कोई elements नहीं हैं

Answer: B. ℚ is countable / ℚ countable है

15. The set of real numbers ℝ is:
Real numbers ℝ का set क्या है?

A. Finite / Finite
B. Countable / Countable
C. Empty / Empty
D. Uncountable / Uncountable

Answer: D. Uncountable / Uncountable

16. According to Cantor's theorem, for any set A:
Cantor's theorem के अनुसार किसी भी set A के लिए:

A. |P(A)| = |A| always / |P(A)| = |A| हमेशा
B. |P(A)| < |A| always / |P(A)| < |A| हमेशा
C. |P(A)| > |A| / |P(A)| > |A|
D. P(A) is always empty / P(A) हमेशा empty होता है

Answer: C. |P(A)| > |A| / |P(A)| > |A|

17. If A has 3 elements and B has 4 elements, then the number of ordered pairs in A × B is:
यदि A में 3 elements और B में 4 elements हैं, तो A × B में ordered pairs की संख्या कितनी होगी?

A. 12 / 12
B. 7 / 7
C. 8 / 8
D. 16 / 16

Answer: A. 12 / 12

18. The equivalence classes of an equivalence relation form a:
किसी equivalence relation की equivalence classes किसका निर्माण करती हैं?

A. Cartesian product / Cartesian product
B. Partition of the set / Set का partition
C. Power set only / केवल power set
D. Empty set / Empty set

Answer: B. Partition of the set / Set का partition

19. Which relation on ℝ is a total order relation?
ℝ पर निम्नलिखित में से कौन-सा relation total order relation है?

A. xRy if x + y = 0 / xRy यदि x + y = 0
B. xRy if x ≠ y / xRy यदि x ≠ y
C. xRy if x ≤ y / xRy यदि x ≤ y
D. xRy if x + y > 0 / xRy यदि x + y > 0

Answer: C. xRy if x ≤ y / xRy यदि x ≤ y

20. The divisibility relation on the set of positive integers is:
Positive integers के set पर divisibility relation क्या है?

A. A total order but not a partial order / Total order लेकिन partial order नहीं
B. A partial order but not a total order / Partial order लेकिन total order नहीं
C. Neither a partial order nor a total order / न partial order और न total order
D. A symmetric relation / Symmetric relation

Answer: B. A partial order but not a total order / Partial order लेकिन total order नहीं

21. If A = {1, 2} and B = {a, b, c}, then the number of elements in A × B is:
यदि A = {1, 2} और B = {a, b, c} है, तो A × B में elements की संख्या कितनी है?

A. 5 / 5
B. 6 / 6
C. 7 / 7
D. 8 / 8

Answer: B. 6 / 6

22. If (a, b) = (c, d), then which condition is necessarily true?
यदि (a, b) = (c, d) है, तो निम्नलिखित में से कौन-सी condition आवश्यक रूप से सही है?

A. a = c and b = d / a = c और b = d
B. a = d and b = c / a = d और b = c
C. a + b = c + d only / केवल a + b = c + d
D. ab = cd only / केवल ab = cd

Answer: A. a = c and b = d / a = c और b = d

23. If A is a set with n elements, then the number of ordered pairs in A × A is:
यदि A में n elements हैं, तो A × A में ordered pairs की संख्या कितनी होगी?

A. n / n
B. 2n / 2n
C. n² / n²
D. n + 1 / n + 1

Answer: C. n² / n²

24. A relation R on a set A is reflexive if:
किसी set A पर relation R reflexive कहलाता है, यदि:

A. aRb for every a, b ∈ A / प्रत्येक a, b ∈ A के लिए aRb
B. aRa for every a ∈ A / प्रत्येक a ∈ A के लिए aRa
C. aRb implies bRa / aRb से bRa प्राप्त हो
D. aRb and bRc imply aRc / aRb और bRc से aRc प्राप्त हो

Answer: B. aRa for every a ∈ A / प्रत्येक a ∈ A के लिए aRa

25. A relation R on A is symmetric if:
A पर relation R symmetric कहलाता है, यदि:

A. aRa for every a ∈ A / प्रत्येक a ∈ A के लिए aRa
B. aRb implies a = b / aRb से a = b
C. aRb implies bRa / aRb से bRa
D. aRb and bRc imply aRc / aRb और bRc से aRc

Answer: C. aRb implies bRa / aRb से bRa

26. A relation R is transitive if:
कोई relation R transitive कहलाता है, यदि:

A. aRb implies bRa / aRb से bRa
B. aRa for every a / प्रत्येक a के लिए aRa
C. aRb implies a = b / aRb से a = b
D. aRb and bRc imply aRc / aRb और bRc से aRc

Answer: D. aRb and bRc imply aRc / aRb और bRc से aRc

27. Which of the following relations on integers is an equivalence relation?
Integers पर निम्नलिखित में से कौन-सा relation equivalence relation है?

A. aRb if a − b is divisible by 3 / aRb यदि a − b, 3 से विभाज्य है
B. aRb if a < b / aRb यदि a < b
C. aRb if a = b + 1 / aRb यदि a = b + 1
D. aRb if a > b / aRb यदि a > b

Answer: A. aRb if a − b is divisible by 3 / aRb यदि a − b, 3 से विभाज्य है

28. Two elements of a set are equivalent under an equivalence relation when they:
किसी equivalence relation के अंतर्गत किसी set के दो elements equivalent कब कहलाते हैं?

A. Are necessarily equal / आवश्यक रूप से equal हों
B. Are related by the equivalence relation / equivalence relation द्वारा related हों
C. Belong to different sets / अलग-अलग sets से belong करते हों
D. Are both empty / दोनों empty हों

Answer: B. Are related by the equivalence relation / equivalence relation द्वारा related हों

29. In a partition of a set, any two distinct blocks are:
किसी set के partition में कोई भी दो distinct blocks कैसे होते हैं?

A. Equal / Equal
B. Overlapping / Overlapping
C. Pairwise disjoint / Pairwise disjoint
D. Always infinite / हमेशा infinite

Answer: C. Pairwise disjoint / Pairwise disjoint

30. The union of all blocks of a partition of A is:
A के partition के सभी blocks का union क्या होता है?

A. Empty set / Empty set
B. A itself / स्वयं A
C. A proper subset of A / A का proper subset
D. Universal set only / केवल universal set

Answer: B. A itself / स्वयं A

31. A partial order relation must be:
किसी partial order relation में निम्नलिखित में से कौन-से गुण होने चाहिए?

A. Reflexive, symmetric and transitive / Reflexive, symmetric और transitive
B. Irreflexive, symmetric and transitive / Irreflexive, symmetric और transitive
C. Reflexive, antisymmetric and transitive / Reflexive, antisymmetric और transitive
D. Symmetric and antisymmetric only / केवल symmetric और antisymmetric

Answer: C. Reflexive, antisymmetric and transitive / Reflexive, antisymmetric और transitive

32. In a partial order, if aRb and bRa, then:
किसी partial order में यदि aRb और bRa है, तो:

A. a = b / a = b
B. a ≠ b / a ≠ b
C. aRb is false / aRb false है
D. bRa is false / bRa false है

Answer: A. a = b / a = b

33. A poset is a set together with:
Poset किसका combination होता है?

A. An equivalence relation / एक equivalence relation
B. A partial order relation / एक partial order relation
C. A symmetric relation only / केवल एक symmetric relation
D. A function only / केवल एक function

Answer: B. A partial order relation / एक partial order relation

34. In a total order, every pair of elements is:
Total order में प्रत्येक pair of elements क्या होता है?

A. Equal / Equal
B. Unrelated / Unrelated
C. Comparable / Comparable
D. Disjoint / Disjoint

Answer: C. Comparable / Comparable

35. The domain of a function f: A → B is:
Function f: A → B का domain क्या होता है?

A. A / A
B. B / B
C. A × B / A × B
D. A ∪ B / A ∪ B

Answer: A. A / A

36. The codomain of a function f: A → B is:
Function f: A → B का codomain क्या होता है?

A. A / A
B. B / B
C. A ∩ B / A ∩ B
D. A − B / A − B

Answer: B. B / B

37. The range of a function f: A → B is:
Function f: A → B का range क्या होता है?

A. The set of elements of B that are images of elements of A / B के वे elements जो A के elements की images हैं
B. The entire set A / पूरा set A
C. The Cartesian product A × B / Cartesian product A × B
D. The empty set always / हमेशा empty set

Answer: A. The set of elements of B that are images of elements of A / B के वे elements जो A के elements की images हैं

38. If f: A → B and g: B → C, then g ∘ f maps:
यदि f: A → B और g: B → C है, तो g ∘ f किससे किसमें map करता है?

A. B → A / B → A
B. C → A / C → A
C. A → C / A → C
D. C → B / C → B

Answer: C. A → C / A → C

39. If f: A → B is invertible, then f must be:
यदि f: A → B invertible है, तो f का होना आवश्यक है:

A. Constant / Constant
B. Bijective / Bijective
C. Only surjective / केवल surjective
D. Only injective / केवल injective

Answer: B. Bijective / Bijective

40. If a finite set A has n elements, then the number of its subsets is:
यदि finite set A में n elements हैं, तो उसके subsets की संख्या कितनी होगी?

A. n / n
B. n² / n²
C. 2n / 2n
D. 2ⁿ / 2ⁿ

Answer: D. 2ⁿ / 2ⁿ

41. If A = {1, 2} and B = {a, b}, which of the following is an element of A × B?
यदि A = {1, 2} और B = {a, b} है, तो निम्नलिखित में से कौन A × B का एक element है?

A. (a, 1) / (a, 1)
B. (1, a) / (1, a)
C. {1, a} / {1, a}
D. (1, 2) / (1, 2)

Answer: B. (1, a) / (1, a)

42. If either A or B is empty, then A × B is:
यदि A या B में से कोई एक empty set है, तो A × B क्या होगा?

A. A / A
B. B / B
C. A ∪ B / A ∪ B
D. ∅ / ∅

Answer: D. ∅ / ∅

43. For any non-empty sets A and B, A × B = B × A generally:
किसी भी non-empty sets A और B के लिए सामान्यतः A × B = B × A क्या होता है?

A. Always true / हमेशा true
B. Always false / हमेशा false
C. True only when A = B / केवल जब A = B हो तब true
D. True only when A and B are finite / केवल जब A और B finite हों तब true

Answer: C. True only when A = B / केवल जब A = B हो तब true

44. A relation R on A is antisymmetric if aRb and bRa imply:
किसी set A पर relation R antisymmetric है, यदि aRb और bRa से क्या प्राप्त होता है?

A. a = b / a = b
B. a ≠ b / a ≠ b
C. aRb is false / aRb false है
D. bRa is false / bRa false है

Answer: A. a = b / a = b

45. The relation “is equal to” on any set is:
किसी भी set पर “is equal to” relation क्या होता है?

A. Only symmetric / केवल symmetric
B. Only transitive / केवल transitive
C. An equivalence relation / एक equivalence relation
D. Neither reflexive nor transitive / न reflexive न transitive

Answer: C. An equivalence relation / एक equivalence relation

46. The relation “less than” (<) on the set of real numbers is:
Real numbers के set पर “less than” (<) relation क्या है?

A. Reflexive / Reflexive
B. Irreflexive / Irreflexive
C. Symmetric / Symmetric
D. Antisymmetric and reflexive / Antisymmetric और reflexive

Answer: B. Irreflexive / Irreflexive

47. The equivalence class of an element a under relation R is denoted by:
Relation R के अंतर्गत element a की equivalence class को किससे denote किया जाता है?

A. [a] / [a]
B. (a) / (a)
C. {a} only / केवल {a}
D. R(a) / R(a)

Answer: A. [a] / [a]

48. If R is an equivalence relation on A and b ∈ [a], then:
यदि R, A पर equivalence relation है और b ∈ [a], तो:

A. a and b are not related / a और b related नहीं हैं
B. aRb / aRb
C. bRa is impossible / bRa impossible है
D. a = b must hold / आवश्यक रूप से a = b होगा

Answer: B. aRb / aRb

49. Two equivalence classes of the same equivalence relation are either:
एक ही equivalence relation की दो equivalence classes या तो क्या होती हैं?

A. Always overlapping / हमेशा overlapping
B. Always equal / हमेशा equal
C. Disjoint or identical / Disjoint या identical
D. Always empty / हमेशा empty

Answer: C. Disjoint or identical / Disjoint या identical

50. Every equivalence relation on a set determines a:
किसी set पर प्रत्येक equivalence relation किसका निर्धारण करता है?

A. Partition of the set / Set का partition
B. Cartesian product only / केवल Cartesian product
C. Single element set / Single element set
D. Empty relation only / केवल empty relation

Answer: A. Partition of the set / Set का partition

51. Conversely, every partition of a set determines:
इसके विपरीत, किसी set का प्रत्येक partition किसे निर्धारित करता है?

A. A function / एक function
B. An equivalence relation / एक equivalence relation
C. A total order only / केवल total order
D. An empty set / एक empty set

Answer: B. An equivalence relation / एक equivalence relation

52. The identity function on a set A is defined by:
किसी set A पर identity function किस प्रकार defined होता है?

A. f(x) = 0 / f(x) = 0
B. f(x) = 1 / f(x) = 1
C. f(x) = x / f(x) = x
D. f(x) = −x / f(x) = −x

Answer: C. f(x) = x / f(x) = x

53. If f: A → B and f(A) = B, then f is:
यदि f: A → B और f(A) = B है, तो f क्या है?

A. Injective necessarily / आवश्यक रूप से injective
B. Constant necessarily / आवश्यक रूप से constant
C. Surjective / Surjective
D. Neither injective nor surjective / न injective न surjective

Answer: C. Surjective / Surjective

54. If f(a) = f(b) implies a = b, then f is:
यदि f(a) = f(b) से a = b प्राप्त होता है, तो f क्या है?

A. Surjective / Surjective
B. Injective / Injective
C. Constant / Constant
D. Identity necessarily / आवश्यक रूप से identity

Answer: B. Injective / Injective

55. For functions f and g for which both compositions are defined, composition of functions is generally:
ऐसे functions f और g जिनके लिए दोनों compositions defined हैं, उनके लिए composition of functions सामान्यतः क्या होता है?

A. Commutative / Commutative
B. Always constant / हमेशा constant
C. Not commutative / Not commutative
D. Always symmetric / हमेशा symmetric

Answer: C. Not commutative / Not commutative

56. If f: A → B is bijective, then f⁻¹ is a function from:
यदि f: A → B bijective है, तो f⁻¹ किससे किसमें function होता है?

A. A → B / A → B
B. B → A / B → A
C. A → A / A → A
D. B → B / B → B

Answer: B. B → A / B → A

57. For an invertible function f, which identity is correct?
किसी invertible function f के लिए कौन-सी identity सही है?

A. f⁻¹ ∘ f = I_A and f ∘ f⁻¹ = I_B / f⁻¹ ∘ f = I_A और f ∘ f⁻¹ = I_B
B. f⁻¹ ∘ f = 0 / f⁻¹ ∘ f = 0
C. f ∘ f⁻¹ = f / f ∘ f⁻¹ = f
D. f⁻¹ = f always / f⁻¹ = f हमेशा

Answer: A. f⁻¹ ∘ f = I_A and f ∘ f⁻¹ = I_B / f⁻¹ ∘ f = I_A और f ∘ f⁻¹ = I_B

58. A finite set is always:
कोई finite set हमेशा क्या होता है?

A. Uncountable / Uncountable
B. Countable / Countable
C. Infinite / Infinite
D. Empty / Empty

Answer: B. Countable / Countable

59. Which of the following sets is countably infinite?
निम्नलिखित में से कौन-सा set countably infinite है?

A. Natural numbers ℕ / Natural numbers ℕ
B. Real numbers ℝ / Real numbers ℝ
C. Interval (0,1) / Interval (0,1)
D. Power set P(ℕ) / Power set P(ℕ)

Answer: A. Natural numbers ℕ / Natural numbers ℕ

60. According to Cantor's theorem, the power set P(A) always has:
Cantor's theorem के अनुसार power set P(A) में हमेशा क्या होता है?

A. Fewer elements than A / A से कम elements
B. The same number of elements as A / A के बराबर elements
C. Strictly more elements than A / A से strictly अधिक elements
D. No elements / कोई elements नहीं

Answer: C. Strictly more elements than A / A से strictly अधिक elements

61. If A has 4 elements and B has 5 elements, then the number of elements in A × B is:
यदि A में 4 elements और B में 5 elements हैं, तो A × B में elements की संख्या कितनी होगी?

A. 9 / 9
B. 20 / 20
C. 16 / 16
D. 25 / 25

Answer: B. 20 / 20

62. If A × B = ∅, then which statement is necessarily true?
यदि A × B = ∅ है, तो निम्नलिखित में से कौन-सा कथन आवश्यक रूप से सही है?

A. A = B / A = B
B. A = B = ∅ / A = B = ∅
C. At least one of A and B is empty / A और B में से कम-से-कम एक empty है
D. Both A and B are infinite / A और B दोनों infinite हैं

Answer: C. At least one of A and B is empty / A और B में से कम-से-कम एक empty है

63. If R is reflexive on A, then for every a ∈ A:
यदि R, A पर reflexive है, तो प्रत्येक a ∈ A के लिए:

A. aRa / aRa
B. aR a is false / aRa false है
C. aRb for every b / प्रत्येक b के लिए aRb
D. bRa for every b / प्रत्येक b के लिए bRa

Answer: A. aRa / aRa

64. If a relation R is both symmetric and antisymmetric, then aRb and bRa imply:
यदि कोई relation R symmetric और antisymmetric दोनों है, तो aRb और bRa से क्या प्राप्त होगा?

A. a < b / a < b
B. a = b / a = b
C. a > b / a > b
D. a ≠ b / a ≠ b

Answer: B. a = b / a = b

65. Which of the following is NOT required for an equivalence relation?
निम्नलिखित में से कौन-सा गुण equivalence relation के लिए आवश्यक नहीं है?

A. Reflexivity / Reflexivity
B. Symmetry / Symmetry
C. Transitivity / Transitivity
D. Antisymmetry / Antisymmetry

Answer: D. Antisymmetry / Antisymmetry

66. The equivalence class [a] consists of:
Equivalence class [a] में क्या शामिल होता है?

A. All elements b such that aRb / वे सभी elements b जिनके लिए aRb
B. Only the element a / केवल element a
C. All elements not related to a / a से related नहीं होने वाले सभी elements
D. All elements outside the set / Set के बाहर के सभी elements

Answer: A. All elements b such that aRb / वे सभी elements b जिनके लिए aRb

67. A partition of a non-empty set cannot contain:
किसी non-empty set के partition में क्या नहीं हो सकता?

A. Two or more blocks / दो या अधिक blocks
B. Non-empty blocks / Non-empty blocks
C. An empty block / एक empty block
D. Pairwise disjoint blocks / Pairwise disjoint blocks

Answer: C. An empty block / एक empty block

68. The relation ≤ on the set of integers is:
Integers के set पर relation ≤ क्या है?

A. An equivalence relation / एक equivalence relation
B. A total order relation / एक total order relation
C. A symmetric relation / एक symmetric relation
D. Neither reflexive nor transitive / न reflexive न transitive

Answer: B. A total order relation / एक total order relation

69. The divisibility relation | on positive integers is reflexive because:
Positive integers पर divisibility relation | reflexive है क्योंकि:

A. a divides b for every a, b / प्रत्येक a, b के लिए a, b को divide करता है
B. a divides a for every positive integer a / प्रत्येक positive integer a के लिए a, a को divide करता है
C. a divides 1 for every a / प्रत्येक a के लिए a, 1 को divide करता है
D. 1 divides a only / केवल 1, a को divide करता है

Answer: B. a divides a for every positive integer a / प्रत्येक positive integer a के लिए a, a को divide करता है

70. A function f: A → B is called into if:
Function f: A → B को into function कब कहा जाता है?

A. Range of f is a proper subset of B / f का range, B का proper subset हो
B. Range of f equals B / f का range B के बराबर हो
C. Every element of B has a pre-image / B के प्रत्येक element की pre-image हो
D. A and B are empty / A और B empty हों

Answer: A. Range of f is a proper subset of B / f का range, B का proper subset हो

71. A constant function maps every element of its domain to:
Constant function अपने domain के प्रत्येक element को किसमें map करता है?

A. A different element / अलग-अलग element
B. Every element of codomain / Codomain के प्रत्येक element
C. The same element of the codomain / Codomain के एक ही element में
D. No element / किसी element में नहीं

Answer: C. The same element of the codomain / Codomain के एक ही element में

72. If f: A → B is injective and A, B are finite sets, then:
यदि f: A → B injective है और A, B finite sets हैं, तो:

A. |A| ≥ |B| / |A| ≥ |B|
B. |A| ≤ |B| / |A| ≤ |B|
C. |A| = |B| always / हमेशा |A| = |B|
D. |A| > |B| always / हमेशा |A| > |B|

Answer: B. |A| ≤ |B| / |A| ≤ |B|

73. If f: A → B is surjective and A, B are finite sets, then:
यदि f: A → B surjective है और A, B finite sets हैं, तो:

A. |A| ≥ |B| / |A| ≥ |B|
B. |A| < |B| always / हमेशा |A| < |B|
C. |A| = 0 / |A| = 0
D. |A| ≤ |B| always / हमेशा |A| ≤ |B|

Answer: A. |A| ≥ |B| / |A| ≥ |B|

74. If f and g are injective functions, then g ∘ f is:
यदि f और g injective functions हैं, तो g ∘ f क्या होगा?

A. Always constant / हमेशा constant
B. Always surjective / हमेशा surjective
C. Injective / Injective
D. Never a function / कभी function नहीं

Answer: C. Injective / Injective

75. If f and g are surjective functions, then g ∘ f is:
यदि f और g surjective functions हैं, तो g ∘ f क्या होगा?

A. Injective / Injective
B. Surjective / Surjective
C. Constant / Constant
D. Identity necessarily / आवश्यक रूप से identity

Answer: B. Surjective / Surjective

76. If f: A → B is bijective, then the cardinalities of A and B are:
यदि f: A → B bijective है, तो A और B की cardinalities कैसी होंगी?

A. |A| < |B| / |A| < |B|
B. |A| > |B| / |A| > |B|
C. |A| = |B| / |A| = |B|
D. Cannot be compared / तुलना नहीं की जा सकती

Answer: C. |A| = |B| / |A| = |B|

77. Two sets A and B have the same cardinality if there exists a:
दो sets A और B की cardinality समान होती है यदि उनके बीच एक क्या मौजूद हो?

A. Bijective function / Bijective function
B. Constant function / Constant function
C. Empty relation / Empty relation
D. Partial order only / केवल partial order

Answer: A. Bijective function / Bijective function

78. Which of the following is an uncountable set?
निम्नलिखित में से कौन-सा uncountable set है?

A. Natural numbers / Natural numbers
B. Integers / Integers
C. Rational numbers / Rational numbers
D. Real numbers / Real numbers

Answer: D. Real numbers / Real numbers

79. The set of all finite subsets of a countable set is:
किसी countable set के सभी finite subsets का set क्या होता है?

A. Always uncountable / हमेशा uncountable
B. Countable / Countable
C. Always empty / हमेशा empty
D. Always finite / हमेशा finite

Answer: B. Countable / Countable

80. If A is a finite set with n elements, then the cardinality of its power set P(A) is:
यदि A एक finite set है जिसमें n elements हैं, तो उसके power set P(A) की cardinality कितनी होगी?

A. n / n
B. n² / n²
C. 2ⁿ / 2ⁿ
D. 2n / 2n

Answer: C. 2ⁿ / 2ⁿ

81. If A has 2 elements and B has 3 elements, then the number of subsets of A × B is:
यदि A में 2 elements और B में 3 elements हैं, तो A × B के subsets की संख्या कितनी होगी?

A. 32 / 32
B. 64 / 64
C. 16 / 16
D. 8 / 8

Answer: B. 64 / 64

82. If R is an equivalence relation on A, then the equivalence classes of R:
यदि R, A पर एक equivalence relation है, तो R की equivalence classes:

A. Form a partition of A / A का partition बनाती हैं
B. Always overlap partially / हमेशा partially overlap करती हैं
C. Are always singletons / हमेशा singletons होती हैं
D. Do not contain any element of A / A का कोई element contain नहीं करती हैं

Answer: A. Form a partition of A / A का partition बनाती हैं

83. Which relation on the set of integers is an equivalence relation?
Integers के set पर निम्नलिखित में से कौन-सा equivalence relation है?

A. aRb if a < b / aRb यदि a < b
B. aRb if a = b + 1 / aRb यदि a = b + 1
C. aRb if a − b is even / aRb यदि a − b even है
D. aRb if a > b / aRb यदि a > b

Answer: C. aRb if a − b is even / aRb यदि a − b even है

84. In a partition of a set A, every element of A belongs to:
किसी set A के partition में A का प्रत्येक element किससे belong करता है?

A. No block / किसी block से नहीं
B. Exactly one block / Exactly one block से
C. At least two blocks / कम-से-कम दो blocks से
D. Every block / प्रत्येक block से

Answer: B. Exactly one block / Exactly one block से

85. Which of the following relations is a partial order on the power set P(A)?
निम्नलिखित में से कौन-सा relation power set P(A) पर partial order है?

A. Equality only / केवल equality
B. Symmetry relation / Symmetry relation
C. Set inclusion ⊆ / Set inclusion ⊆
D. Inequality ≠ / Inequality ≠

Answer: C. Set inclusion ⊆ / Set inclusion ⊆

86. The relation ⊆ on P(A) is reflexive because:
P(A) पर relation ⊆ reflexive है क्योंकि:

A. Every set is a subset of itself / प्रत्येक set स्वयं का subset होता है
B. No set is a subset of itself / कोई set स्वयं का subset नहीं होता
C. All sets are equal / सभी sets equal होते हैं
D. A is always empty / A हमेशा empty होता है

Answer: A. Every set is a subset of itself / प्रत्येक set स्वयं का subset होता है

87. If f: A → B is both injective and surjective, then every element of B has:
यदि f: A → B injective और surjective दोनों है, तो B के प्रत्येक element की:

A. No pre-image / कोई pre-image नहीं
B. At least two pre-images / कम-से-कम दो pre-images
C. Exactly one pre-image / Exactly one pre-image
D. Infinitely many pre-images / Infinitely many pre-images

Answer: C. Exactly one pre-image / Exactly one pre-image

88. If f: A → B is invertible, then f⁻¹ ∘ f is:
यदि f: A → B invertible है, तो f⁻¹ ∘ f क्या होगा?

A. f / f
B. I_A / I_A
C. I_B / I_B
D. 0 / 0

Answer: B. I_A / I_A

89. If f: A → B and g: B → C are functions, then f ∘ g is defined only when:
यदि f: A → B और g: B → C functions हैं, तो f ∘ g कब defined होता है?

A. The codomain of g matches the domain of f / g का codomain, f के domain से match करे
B. The domain of g matches the domain of f / g का domain, f के domain से match करे
C. A = C always / हमेशा A = C हो
D. A = B = C always / हमेशा A = B = C हो

Answer: A. The codomain of g matches the domain of f / g का codomain, f के domain से match करे

90. If f: A → B and g: B → C, then the domain of g ∘ f is:
यदि f: A → B और g: B → C है, तो g ∘ f का domain क्या है?

A. B / B
B. C / C
C. A / A
D. A × C / A × C

Answer: C. A / A

91. A set that can be put into one-to-one correspondence with the natural numbers is called:
जिस set को natural numbers के साथ one-to-one correspondence में रखा जा सकता है, उसे क्या कहते हैं?

A. Finite set only / केवल finite set
B. Countably infinite set / Countably infinite set
C. Uncountable set / Uncountable set
D. Empty set / Empty set

Answer: B. Countably infinite set / Countably infinite set

92. Which of the following sets is countable?
निम्नलिखित में से कौन-सा set countable है?

A. Integers ℤ / Integers ℤ
B. Real numbers ℝ / Real numbers ℝ
C. Irrational numbers / Irrational numbers
D. The interval (0,1) of real numbers / Real numbers का interval (0,1)

Answer: A. Integers ℤ / Integers ℤ

93. The set of rational numbers ℚ is:
Rational numbers ℚ का set क्या है?

A. Finite / Finite
B. Uncountable / Uncountable
C. Countably infinite / Countably infinite
D. Empty / Empty

Answer: C. Countably infinite / Countably infinite

94. Cantor's diagonal argument is used to prove that:
Cantor का diagonal argument किस बात को सिद्ध करने के लिए प्रयोग किया जाता है?

A. Natural numbers are finite / Natural numbers finite हैं
B. Integers are uncountable / Integers uncountable हैं
C. Rational numbers are uncountable / Rational numbers uncountable हैं
D. Real numbers are uncountable / Real numbers uncountable हैं

Answer: D. Real numbers are uncountable / Real numbers uncountable हैं

95. Cantor's theorem states that for every set A:
Cantor's theorem के अनुसार प्रत्येक set A के लिए:

A. A and P(A) always have the same cardinality / A और P(A) की cardinality हमेशा समान होती है
B. There is no bijection from A onto P(A) / A से P(A) पर कोई bijection नहीं होता है
C. P(A) is always empty / P(A) हमेशा empty होता है
D. A is always uncountable / A हमेशा uncountable होता है

Answer: B. There is no bijection from A onto P(A) / A से P(A) पर कोई bijection नहीं होता है

96. If A is a set with 5 elements, then the cardinality of P(A) is:
यदि A में 5 elements हैं, तो P(A) की cardinality कितनी होगी?

A. 10 / 10
B. 16 / 16
C. 25 / 25
D. 32 / 32

Answer: D. 32 / 32

97. If A is countably infinite and B is finite, then A ∪ B is:
यदि A countably infinite है और B finite है, तो A ∪ B क्या होगा?

A. Countably infinite / Countably infinite
B. Finite / Finite
C. Uncountable / Uncountable
D. Empty / Empty

Answer: A. Countably infinite / Countably infinite

98. If f: A → B is bijective, then f⁻¹ is:
यदि f: A → B bijective है, तो f⁻¹ क्या होगा?

A. Neither injective nor surjective / न injective न surjective
B. Only constant / केवल constant
C. Bijective / Bijective
D. Only into / केवल into

Answer: C. Bijective / Bijective

99. If f and g are bijective functions, then g ∘ f is:
यदि f और g bijective functions हैं, तो g ∘ f क्या होगा?

A. Constant / Constant
B. Bijective / Bijective
C. Neither injective nor surjective / न injective न surjective
D. Always an empty function / हमेशा empty function

Answer: B. Bijective / Bijective

100. Which statement correctly describes the relationship between an equivalence relation and a partition?
Equivalence relation और partition के बीच संबंध को कौन-सा कथन सही रूप से बताता है?

A. Every partition determines an equivalence relation, and every equivalence relation determines a partition / प्रत्येक partition एक equivalence relation निर्धारित करता है और प्रत्येक equivalence relation एक partition निर्धारित करता है
B. They are completely unrelated / दोनों का आपस में कोई संबंध नहीं है
C. Every equivalence relation is a total order / प्रत्येक equivalence relation total order है
D. Every partition is a function / प्रत्येक partition एक function है

Answer: A. Every partition determines an equivalence relation, and every equivalence relation determines a partition / प्रत्येक partition एक equivalence relation निर्धारित करता है और प्रत्येक equivalence relation एक partition निर्धारित करता है