1. In a survey of 85 people it is found that 31 like to drink milk 43 like coffee and 39 like tea.Also 13 like both milk and tea, 15 like milk and coffee, 20 like tea and coffee and 12 like none of the three drinks. Find the number of people who like all the three drinks.
Correct : B. 8
2. Find the negation of the proposition: “Michael’s PC runs Linux”
Correct : C. Both a and b
3. A proof that begins by asserting a claim and proceeds to show that the claim cannot be true is by
Correct : B. Contradiction
4. A proof that proceeds by showing the existence of something desired is by
Correct : A. Induction
5. Proofs by contradiction
Correct : C. start by assuming the opposite of what is to be proven
6. Induction is a
Correct : D. Proof method
7. ^ denotes
Correct : B. AND
8. ~ denotes
Correct : D. negation
9. Quantifiers variables
Correct : C. give values to
10. A validity-maintaining procedure for deriving sentences in logic from other sentences is
Correct : C. Inference rule
11. Inference rules maintain
Correct : B. validity
12. A validity-maintaining procedure for deriving sentences in logic from other sentences is a
Correct : C. Inference rule
13. If A and B be sets and AC and Bc denote the complements of the sets A and B, then set (A —
B) ∪ (B — A) ∪ (A ∩ B) is equal to
Correct : C. A ∪ B
14. Number of proper subsets of a set of order three
Correct : B. 6
15. If A be a finite set of size n, then number of elements in the power set of A x A is
Correct : B. 2n^2
16. Which of the following set (s) are empty ?
Correct : B. {x : x ≠ x}
17. n a Venn diagram , the overlap between two circles represents:
Correct : B. the intersection of two sets
18. Which of these subsets are equal: A = {r.t.s} B = {s,t,r,s} C = {t,s,t,r} D = {s,r,s,t}
Correct : D. all are equal
19. Determine the total number of subsets of the following set: {h,i, j, k, l, m, n}
Correct : A. 128
20. If B is a Boolean Algebra, then which of the following is true
Correct : B. B is a finite, complemented and distributive lattice
21. The statement ( p^q) _ p is a
Correct : C. tautology
22. 1. Let m = “Juan is a math major,” c = “Juan is a computer science major,” g = “Juan’s girlfriend is a literature major,” h = “Juan’s girlfriend has read Hamlet,” and t = “Juan’s girlfriend has read The Tempest.” Which of the following expresses the statement “Juan is a computer science major and a math major, but his girlfriend is a literature major who hasn’t read both The Tempest and Hamlet.”
Correct : C. c ∧ m ∧ g ∧
(∼h ∨ ∼t)
23. The truth table for (p ∨ q) ∨ (p ∧ r) is the same as the
truth table for
Correct : D. p V q
24. Consider the statement, “Either −2 ≤ x ≤ −1 or 1 ≤ x ≤
2.” The negation of this statement is
Correct : A. x < −2 or 2 < x
or −1 < x < 1
25. Which of the following statements is FALSE:
Correct : A. (P ∧ Q) ∨ (∼P
∧ Q) ∨ (P ∧
∼Q) is equal to
∼Q ∧ ∼P
26. anya is older than Eric. Cliff is older than Tanya. Eric is older than Cliff. If the first two statements are true, the third statement is
Correct : B. FALSE
27. Blueberries cost more than strawberries. Blueberries cost less than raspberries.Raspberries cost more than both strawberries and blueberries. If the first two statements are true, the third statement is
Correct : A. TRUE
28. All the trees in the park are flowering trees. Some of the trees in the park are dogwoods. All dogwoods in the park are flowering trees. If the first two statements are true, the third statement is
Correct : A. TRUE
29. Mara runs faster than Gail.Lily runs faster than Mara.Gail runs faster than Lily.If the first two statements are true, the third statement is
Correct : B. FALSE
30. Apartments in the Riverdale Manor cost less than apartments in The Gaslight Commons.Apartments in the Livingston Gate cost more than apartments in the The Gaslight Commons.Of the three apartment buildings, the Livingston Gate costs the most.If the first two statements are true, the third statement is
Correct : A. TRUE
31. The power set of the set {ϕ} is
Correct : B. {ϕ, {ϕ}}
32. Fact 1:All dogs like to run.Fact 2:Some dogs like to swim.Fact 3:Some dogs look like their masters.If the first three statements are facts, which of the following statements must also be a fact? I:All dogs who like to swim look like their masters. II:Dogs who like to swim also like to run.III:Dogs who like to run do not look like their masters.
Correct : B. II only
33. Fact 1: Jessica has four children
Fact 2: Two of the children have blue eyes and two of the children have brown eyes.
Fact 3: Half of the children are girls.
If the first three statements are facts, which of the following statements must also be a fact?
I: At least one girl has blue eyes. II: Two of the children are boys. III: The boys have brown eyes.
Correct : B. II only
34. Fact 1: All drink mixes are beverages. Fact 2: All beverages are drinkable. Fact 3: Some beverages are red.
If the first three statements are facts, which of the following statements must also be a fact?
I: Some drink mixes are red.
II: All beverages are drink mixes.
III: All red drink mixes are drinkable.
Correct : C. III only
35. Fact 1: All chickens are birds. Fact 2: Some chickens are hens. Fact 3: Female birds lay eggs.
If the first three statements are facts, which of the following statements must also be a fact?
I: All birds lay eggs.
II: Some Hens are birds.
III: Some chickens are not hens.
Correct : C. II and III only
36. 100 sportsmen were asked whether they play which game: Cricket, hockey,Football. The results were : 45 play cricket, 38 play hockey, 21 play football, 18 play cricket and hockey, 9 play cricket and football, 4 play football and hockey and 23 play none of these.
Determine the number of sportsmen who play exactly 1game
Correct : A. 54
37. In above Ex. 196 how many players play exactly 2 of the games?
Correct : C. 19
38. A theory of sets was firstly introduced by….
Correct : C. G.Canter
39. the inventor of set defined set as
Correct : C. simple as collection of objects
40. class, groups , collection are synonyms of the term set.
Correct : A. TRUE
41. Define f(n) = n/2 + 1−(−1)n/4 for all n 2 Z. Thus, f: Z
→ Z, Z the set of all integers.Which is correct?
Correct : A. f is a function and is onto but not one-to-one.
42. If f (x) = cos x and g(x) = x3 , then (f o g) (x) is
Correct : D. cos x3
43. Transitivity and irreflexive imply:
Correct : D. Asymmetric
44. If R is a relation “Less Than” from A = {1,2,3,4} to B =
{1,3,5} then RoR-1 is
Correct : C. {(3,3), (3,5), (5,3),
(5,5)}
45. A relation that is reflexive, anti-symmetric and transitive is a
Correct : C. partial order
46. Let f : X →Y and g : Y → Z. Let h = go f : X → Z. Suppose g is one-to-one and onto. Which of the following is FALSE?
Correct : A. If f is one-to-one then h is one-to- one and onto.
47. Domain and Range of the function Y = –v(–2x + 3) is
Correct : D. x=3/2, y=0
48. The relation { (1,2), (1,3), (3,1), (1,1), (3,3), (3,2),
(1,4), (4,2), (3,4)} is
Correct : B. Transitive
49. A partial ordered relation is transitive, reflexive and
Correct : A. Anti-symmetric
50. Find the number of relations from A = {cat, dog, rat} to B = {male , female}
Correct : A. 64
51. The number of distinct relations on a set of 3 elements is
Correct : C. 18
52. How many onto (or surjective) functions are there from an n-element (n => 2) set to a 2-element set?
Correct : C. 2n - 2
53. How many relations are there on a set with n elements that are symmetric and a set with n elements that are reflexive and symmetric?
Correct : D. 2n(n+1)/2 and 2n(n–1)/2
54. The number of functions from an m element set to an n element set is:
Correct : A. mn
55. Let R = {(a, a), (a, b)} be a relation on S = {a, b, c}. Then R is not reflexive and not symmetric.
Correct : A. T
56. Let R = {(a, a), (a, b)} be a relation on S = {a, b, c}. Then R is not transitive.
Correct : B. F
57. The function f : Z → Z given by f (x)= x +1 is a bijection.
Correct : A. T
58. The function f : A → B is injective if whenever f (x)= f (y), where x, y € A, then x = y.
Correct : A. T
59. Let A be a finite set. If f : A → A is injective then it is surjective.
Correct : A. T
60. The union of two equivalence relations on a non- empty set is an equivalence relation.
Correct : B. F
61. If A is a set with 3 elements, how many equivalence relations are there on A? Hint: The set of equivalence classes for a given equivalence relation on A is a partition of the set A.
Correct : B. 5
62. Let f: A → B and g: B→C be functions where A = {1, 2, 3, 4}, B = {1, 2, 3, 4, 5}, and C = {1, 2, 3, 4, 5, 6}, f =
{(1, 2), (2, 3), (3, 2), (4, 5)} and g = {(1, 3), (2, 4), (3,
5), (4, 6), (5, 1)}. Find g o.f (2).
Correct : D. 6
63. A relation R is defined on Z by xRy if 2x +5y = 0(mod7). Then the equivalence class[10] is equal to the equivalence class…
Correct : A. 3
64. Let R be a relation on a set A = {1, 2, 3, 4} given by R =
{(1, 1), (1, 2), (1, 3), (2, 1), (2,2), (2, 3), (3, 1), (3, 2), (3,
3)}. Then the relation is:
Correct : C. symmetric and transitive, but not reflexive.
65. If f (x) = -3x - 5, what is the value of f (2)?
Correct : A. -11
66. If g (x) = 3x² - 2x - 5, what is the value of g (-1)?
Correct : A. -4
67. Which relation is not a function?
Correct : D. {(0,-2), (1,0), (-1,-
3), (0,-1)}
68. Consider the recurrence relation ak = -8ak-1 - 15ak-2 with initial conditions a0 = 0 and a1 = 2. Which of the following is an explicit solution to this recurrence relation?
Correct : A. ak = (-3)k - (-5)k
69. Consider the recurrence relation ak = 6ak-1 - 9ak-2 with initial conditions a0 = 0 and a1 = 2. Which of the following is an explicit solution to this recurrence relation, provided the constants A and B are chosen correctly?
Correct : C. an = A3n + nB3n
70. The binary relation R = {(0, 0), (1, 1)} on A = {0, 1, 2, 3,
} is
Correct : B. Not Reflexive, Symmetric, Transitive
71. Define a binary relation R = {(0, 1), (1, 2), (2, 3), (3, 2),
(2, 0)} on A = {0, 1, 2, 3}. The directed graph (including loops) of the transitive closure of this relation has
Correct : A. 16 arrows
72. Let N+ denote the nonzero natural numbers. Define a binary relation R on N+ × N+ by (m, n)R(s, t) if gcd(m,
n) = gcd(s, t). The binary relation R is
Correct : A. Reflexive, Not Symmetric, Transitive
73. Define a binary relation R on a set A to be anti- reflexive if xRx doesn’t hold for any x 2 A. The number of symmetric, anti-reflexive binary relations on a set of ten elements is
Correct : C. 245
74. Define an equivalence relation R on the positive integers A = {2, 3, 4, . . . , 20} by m R n if the largest prime divisor of m is the same as the largest prime divisor of n. The number of equivalence classes of R is
Correct : A. 8
75. Consider the divides relation, m | n, on the set A = {2, 3, 4, 5, 6, 7, 8, 9, 10}. The cardinality of the covering relation for this partial order relation (i.e., the number of edges in the Hasse diagram) is
Correct : D. 7
76. Let A = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}
and consider the divides relation on A. Let C denote the length of the maximal chain, M the number of maximal elements, and m the number of minimal elements. Which is true?
Correct : A. C = 3, M = 8, m = 6
77. Consider the binary relation R = {(x,y), (x,z), (z,x), (z,y)} on the set {x,y,z}. Which one of the following is TRUE?
Correct : D. R is neither symmetric nor antisymmetric
78. Let S be a set of n elements. The number of ordered pairs in the largest and the smallest equivalence relations on S are:
Correct : A. n and n
79. Which one of the following is the example of nonlinear data structure?
Correct : A. Graph
80. If |A|=5 and |B|=4, then there exists an injective function f: B→A.
Correct : B. F
81. The set of even integers is well-ordered.
Correct : B. F
82. If n is an integer and x is an irrational real number, then nx is irrational.
Correct : B. F
83. In Z7, if [6]x =[3]then x =? .
Correct : D. [4]
84. Which statement represents "all numbers between negative 4 and positive 8" ?
Correct : B. -4 < x < 8
85. Which interval notation represents the set of numbers that are greater than or equal to -1, but are less than 9?
Correct : D. [-1,9)
86. Given the relation D = {(6,4), (8,-1), (x,7), (-3,-6)}. Which of the following values for x will make relation D a function?
Correct : A. -3
87. Which statement is true about the relation shown at the right?
Correct : C. It is not a function because there are multiple x-values for a given y-value.
88. A ball is tossed in the air in such a way that the path of the ball is modeled by the equation
y = -x² + 6x, where y represents the height of the ball in feet and x is the time in seconds. At what time, x, is the ball at its highest point?
Correct : B. 2
89. Let S = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21}. What is the
smallest integer N > 0 such
that for any set of N integers, chosen from S, there must be two distinct integers that
divide each other? IZ
Correct : D. 8
90. If S is a set containing n elements then number of elements in power set of S ,i.e.P(S)
Correct : C. 2n
91. If A and B are two non empty sets then cartesian product of A and B is----------
Correct : C. AΧB={(a,b);aϵA
and bϵB}
92. If set A contains n elements, set B contains m elements then number of elements in AXB is---
Correct : C. m.n
93. If A,B and C are non empty sets then AX(B∩C) is-----.
Correct : B. (AXB)∩(AXC)
94. If A,B and C are non empty sets then AX(BUC) is-----.
Correct : A. (AXB)U(AXC)
95. If R is a relation defined from set A to set B then------
Correct : B. RCAXB
96. If A=(1,2,3} and R on A is defined by R={(1,1),(2,2),(3,3)} then R is…………
Correct : A. Reflexive
97. If A=(1,2,3} and R on A is defined by R={(1,2),(2,1),(1,1),(2,2)} then R is…………
Correct : C. Symmetric and Transitive
98. Which statement is true?
Correct : B. Function can be of the type one many
99. Which statement is true?
Correct : A. Every function is a relation.
100. In-----Relation every element is related with itself.