Review Questions
Question 1 The compound statement P → (P → Q) is false, then the truth
values of P, Q are respectively …….
(a) T, T
(c) T, F
(b) F, T
(d) F, F
Question 2 Let P , Q , R be true , false , false, respectively, which of the
following is true
(a) P ∧ (Q ∧ ¬ R).
(c)
(b) (P → Q) ∧ ¬ R
(d) P ↔ (Q ∨ R).
Q ↔ (P ∧ R).
Question 3 The converse of p → q is
(a) ¬q → ¬p
(c) ¬p → q
(b) ¬p → ¬q
(d) q → p
Question 4 p∧ q is logically equivalent to:
(a) ¬(p → ¬q)
(b) (p → ¬q)
(c) (¬p → ¬q)
(d) (¬p → q)
Question 5 Which of the following statement is not correct?
(a) p ∨ q ≡ q ∨ p
(b) ¬(p ∧ q) ≡ ¬p ∨ ¬q
(c) (p ∨ q) ∨ r ≡ p ∨ (q ∨ r)
(d) Not all of mentioned .
Question 6 (p → q) ∧ (p → r) is logically equivalent to
(a) p → (q ∨ r)
(b) p → (q ∧ r)
(c) p ∧ (q ∨ r)
(d) p ∨ (q ∧ r)
Question 7 Let p: I will get a job, q: I pass the exam, then the statement form:
“I will get a job only if I pass the exam”, in symbolic from is
(a) p ˄ q
(b) q → p
(c)
p∨ q
(d) p → q
Question 8 Let p denote the statement: “Amr is tall”, q: “Amr is handsome”.
Then the negation of the statement “Amr is tall, but not handsome”, in symbolic
form is …..
(a) ¬p ˄ q
(b) ¬p ∨ q
(c)
¬p ∨ ¬q
(d) ¬p ˄ ¬q
1
Question 9 The negation of the statement:
(a)
(c)
For every x ∊ Z there exists a y ∊ Z
There exists an x ∊ Z such that
(b)
for all y ∊ Z, y2 > x.
There exists an x ∊ Z such that
(d)
for all y ∊ Z, y2 ≤ x.
such that y2 > x is ………
There exists a y ∊ Z such that
for every x ∊ Z, y2 > x.
None of the above.
Question 10 “The product of two negative real numbers is not negative.” Is
given by?
(a)
(c)
∃x ∀y((x < 0) ∧ (y < 0) → (xy > 0))
∀x ∃y((x < 0) ∧ (y < 0) ∧ (xy > 0))
(b)
(d)
∃x ∃y((x < 0) ∧ (y < 0) ∧ (xy > 0))
∀x ∀y((x < 0) ∧ (y < 0) → (xy > 0)
Question 11 Which of the arguments is not valid in proving sum of two odd
numbers is not odd.
2n +1 + 2m +1 = 2(n+m+1) hence
true for all
(a)
3 + 3 = 6, hence true for all
(b)
(c)
All of the mentioned
(d) None of the mentioned
Question 12 Which of the following can only be used in disproving the
statements?
(a)
(c)
(b) Contrapositive proofs
(d) Mathematical Induction
Direct proof
Counter Example
Question 13 Decide whether the argument is valid or invalid, and give the form
(of valid or invalid argument) that applies.
If I'm hungry, then I will eat.
(a) Valid; modus ponens
(c)
I'm not hungry. I will not eat.
(b) Invalid; fallacy of the inverse
(d) Invalid; fallacy of the converse
Valid; modus tollens
Question 14 Determine if the argument is valid.
If a number is even, then it is divisible by 2.
If a number is divisible by 8, then it is divisible by 2.
Some numbers are either even or divisible by 8.
Therefore : Some numbers are divisible by 2.
(a)
(b)
Invalid
Valid
Question 15 If n  N , P(n) is a statement such that, if P(k)  P(k+1) for k  N ,
then P(n) is true …….
(a) for all n > 1.
(b) For all n  N .
(c) for all n ≥ 2.
(d) Nothing can be said.
2
Question 16 Suppose that P (n) is a propositional function. Determine for which
positive integers n the statement P (n) must be true if:
P (1) is true; for all positive integers n, if P (n) is true then P (n + 2) is true.
(a) P (2)
(c) P (4)
(b) P (3)
(d) P (6)
Question 17 Suppose that P (n) is a propositional function. Determine for which
positive integers n the statement P (n) must be true if: P (2) is true; for all
positive integers n, if P (n) is true then P (n + 2) is true.
(a) P(1)
(b) P(3)
(c)
(d) Non of the above
P(7)
Question 18 The inequality n! > 2n is true for …..
(a) n  2 , n  N
(b) n  2
nN
(c)
(d) None of these
Question 19 Let the universe of discourse be the set of all integer numbers Z .
If P(x,y) denotes “x + y = 0”. Match the following:
A
B
C
D
xy P(x,y)
xy P(x,y)
xy P(x,y)
xy P(x,y)
Codes: ABCD
(a)
1221
(b)
1122
(d)
(c)
1.
True
2.
False
1112
1121
Question 20 : Let the universe of discourse be the set of all integer numbers Z .
If P(x,y) denotes “x + y = 0”. Match the following:
A
B
C
D
xy P(x,y)
xy P(x,y)
xy P(x,y)
xy P(x,y)
Codes: ABCD
(a)
1221
(b)
1122
(d)
(c)
3
1.
True
2.
False
1112
1121
Question 21 : Let P: I will get a job, Q: I pass the exam, then the statement form:
“I will get a job only if I pass the exam“ , in symbolic from is …..
(a)
P Q
(b)
P ∧Q
(c)
PQ
(d)
Q P
Question 22 : Let p and q be two propositions, then which of the following are
logically equivalent ?
(a)
q  p and p  q
(b)
(p  ( p  q)) and  p  q
(c)
 (A  B) and p  q
(d)
None of the above.
Question 23 : If A = {1, 2, 3, 4, 5, 7, 8, 9} and B = {2, 4, 6, 7, 9} then find the number
of proper subsets of A ∩ B ?
(a)
16
(b)
15
(c)
32
(d)
31
Question 24 : Let us consider the propositional function B(x,y). The negation of:
∀x∃y[B(x, y) ∧ ∀z((z ≠ y) →  B(x, z))] is…..
∃x∀y [B(x, y)  ∃z(z≠y ∧ B(x,z)]
(b) ∃x∀y [B(x, y)  ∃z(z=y ∧ B(x,z)]
(d) non of the above.
∃x∀y [B(x, y)  ∃z(z≠y ∧ B(x,z)]
(a)
(c)
Question 25 : The cardinality of
(a)
(c)
P(P ( )) is ……..
Zero
Two
One
Three
(b)
(d)
Question 26 : Let P(n) : 1  3  5  L (2n1)  3  n2 Then which of the following is true
(a)
P(1) is true.
(b)
P(k)  P(k  1) .
(c)
P(k) is true , P(k 1) is not true
(d)
Both (a) and (b) are true.
Question 27 : Let A and B be any sets. Which of the following statements is True ?
(a)
(c)
  P ( 2,3  ) and   P ( 2,3  ) ,
where P ( 2,3  ) is the power set of
 2 A
 2,3 .
 3 A
implies that  2 , 3   A .
and
(b)
(d)
2  A
 3   A B
implies that  2    3   A  B .
and
None of the above
Question 28 : If A is not a subset of B, then which of the following is correct:
(a)
(c)
x (xA  xB)
∃x (xA ∧  xB)
(b)
x (xB  xA)
None of the above
(d)
4
Question 29 : If f:A→ B is a bijective function and |A| = 6 then which of the
following is not possible
(a)
No of elements in range of f is 6
(b)
|B| = 6
(c)
|B| =8
(d)
|A| = |B|
Question 30 : Which of the following sequences is graphic
(a)
5, 5, 4, 3, 2, 1
(b)
5, 4, 3, 2, 1, 0
(c)
3, 3, 3, 2, 2, 2
(d)
4, 4, 4, 3, 2, 1, 0
Question 31 : Let the universe of discourse be the set of all natural numbers
N = {0, 1, 2, …..}.
A
B
C
D
If P(x,y) denotes “x + y = 0”. Match the following:
xy P(x,y)
xy P(x,y)
xy P(x,y)
xy P(x,y)
1.
True
2.
False
Codes: ABCD
(a)
2121
(b)
(c)
2112
(d)
2122
2222
Question 32 : If A = {1, 2, 3, 4, 5, 7, 8, 9} and B = {2, 4, 6, 7, 9} then find the number
of proper subsets of A ∩ B ?
(a)
15
(b)
16
(c)
31
(d)
32
Question 33 : Let us consider the propositional function B(x,y). The negation of:
(a)
(c)
∀x∃y[B(x, y) ∧ ∀z((z ≠ y) →  B(x, z))] is…..
(b) ∃x∀y [B(x, y)  ∃z(z=y ∧ B(x,z)]
∃x∀y [B(x, y)  ∃z(z≠y ∧ B(x,z)]
∃x∀y [B(x, y)  ∃z(z≠y ∧ B(x,z)]
(d) non of the above.
Question 34 : Let P: I will get a job, Q: I pass the exam, then the statement form:
“I will get a job only if I pass the exam“ , in symbolic from is …..
(a)
PQ
(b)
Q P
(c)
P Q
(d)
P ∧Q
5
Question 35 : Let p and q be two propositions, then which of the following are
logically equivalent ?
(a)
(p  ( p  q)) and  p  q
(b)
 (A  B) and p  q
(c)
q  p and p  q
(d)
None of the above.
Question 36 : Let A and B be any sets. Which of the following statements is True ?
  P ( 2,3  ) and   P ( 2,3  ) ,
where P ( 2,3  ) is the power set of
(a)
 2 A
 3 A
implies that  2 , 3   A .
(b)
 2,3 .
2  A
 3   A B
implies that  2    3   A  B .
(c)
and
and
None of the above
(d)
Question 37 : Let P(n) : 1  3  5  L (2n1)  3  n2 Then which of the following is
true
(a)
P(1) is true.
(b)
P(k)  P(k  1) .
(c)
P(k) is true , P(k 1) is not true
(d)
Both (a) and (b) are true.
Question 38 : The cardinality of
(a)
(c)
P(P ( )) is ……..
Three
One
Two
Zero
(b)
(d)
Question 39 : Which of the following sequences is graphic
(a)
3, 3, 3, 2, 2, 2
(b)
4, 4, 4, 3, 2, 1, 0
(c)
5, 5, 4, 3, 2, 1
(d)
5, 4, 3, 2, 1, 0
Question 40 : If f:A→ B is a bijective function and |A| = 6 then which of the
following is not possible
(a)
No of elements in range of f is 6
(b)
|B| = 6
(c)
|B| =8
(d)
|A| = |B|
Question 41 : If A is not a subset of B, then which of the following is correct:
(a)
(c)
x (xB  xA)
x (xA  xB)
(b)
∃x (xA ∧  xB)
None of the above
(d)
6
Question 42 : Let the universe of discourse be the set of all real numbers R .
If P(x,y) denotes “x y = 1”. Match the following:
A
B
C
D
xy P(x,y)
xy P(x,y)
xy P(x,y)
xy P(x,y)
1.
True
2.
False
Codes: ABCD
(a)
2121
(b)
(c)
2111
(d)
2112
2122
Question 43 : Let p and q be two propositions, then which of the following are not
logically equivalent ?
(a)
q  p and  q   p
(b)
(p  ( p  q)) and  p  q
(c)
 (p  q) and  p   q
(d)
None of the above.
Question 44 : If pq and  q , then ………….
(a)
p
(b)
 p  q
(c)
p
(d)
None of the above.
Fact 1: If 7 is less than 3, then 7 is not a prime number .
Fact 2: 7 is not less than 3.
If the first two statements are facts, which of the following statements must also be
a fact?
I: 7 is a prime number.
II: 3 is a prime number.
(a)
I only
(b) II only
(c)
I and II.
(d) non of the above.
Question 45 :
Question 46 : If A = {1, 2, 3, 4, 5, 7, 8, 9} and B = {2, 4, 6, 7, 9} then find the number
of subsets of A ∩ B ?
(a)
16
(b)
15
(c)
32
(d)
31
Question 47 : For any three sets A, B and C. The set (A – B) – C is equal to the set
(a)
(c)
(A – B) ∩ C
A – (B ∪ C)
(b)
(d)
7
(A ∪ B) – C
(A – B) ∪ C
Question 48 : If  n N ,
P(n) is a statement such that, if P(k) is true, then P(k+1) is true
for
k  N , then P(n) is true
(a)
(c)
For all n > 1.
For all n > 2
(b)
For all n N
(d)
Nothing can be said.
Question 49 : If a relation is reflexive, then all the diagonal entries in the relation
matrix must be________.
(a)
(c)
0
2
(b)
(d)
1
-1
Question 50: Let
R = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4)},
be a relation on the set A= {1, 2, 3,4}, then R is …..
(a)
symmetric
(b)
irreflexive
(c)
antisymmetric
(d)
non of the above.
Question 51: A function f: N → N defined by f(x) = x2 is ……….
(a)
(c)
injective.
surjective.
not injective
not surjective
(b)
(d)
Question 52 : The adjacency matrix of the following graph.
is ……….
(a)
(c)
0
1

1

1
0
1

1

1
1 1 1
0 0 1 
0 0 1

1 0 0
1 1 1
0 0 1 
0 0 1

0 1 0
(b)
(d)
8
0
1

1

1
0
1

1

0
1 1 1
0 0 1 
0 0 1

1 1 0
1 1 1
0 0 1 
0 0 1

1 1 0
Question 53 : Let the universe of discourse be all subsets of the real numbers R .
If P(A,B) denotes “ A  B   ”. Match the following:
A
B
C
D
AB P(A,B)
AB P(A,B)
AB P(A,B)
AB P(A,B)
Codes: ABCD
(a)
1121
(b)
2121
(d)
(c)
1.
True
2.
False
1122
1112
Question 54 : Let p denote the statement: “Amr is tall”, q: “Amr is handsome”. Then
the negation of the statement “Amr is tall, but not handsome”, in symbolic form is …..
(a) ¬ p ˄ q
(b) ¬ p ∨ q
¬p∨ ¬q
(c)
(d) ¬ p ˄ ¬ q
Question 55 : Which of the following is FALSE?
(a)
((x  y) ∧ x)  y
(b)
((x  y) ∧ ( x ∧  y))  y
(c)
(x  ( x v y))
(d)
((x v y)  ( x v  y))
Question 56 : In a survey of 1000 consumers it is found that 720 consumers liked
product A and 450 liked product B.
What is the least number that must have liked both the products?
(a)
70
(b)
170
(c)
270
(d)
Non of the above.
Question 57 : Let us consider the propositional function B(x,y).
(a)
(c)
The logically equivalent of:
∀x∃y[B(x, y) ∧ ∀z((z ≠ y) →  B(x, z))] is…..
∀x∃y [B(x, y)  ∃z(z≠y ∧ B(x,z)]
(b) ∃x∀y [B(x, y)  ∃z(z≠y ∧ B(x,z)]
(d) non of the above.
∀x∃y [ B(x, y) ∧ ∀z(z = y   B(x,z)) ]
9
Question 58 : Consider the following statements :
I. If f is the subset of Z × Z defined by f = {(xy, x − y); x, y ∈ Z}, then f is a function
from Z to Z.
II. If f is the subset of N × N defined by f = {(xy, x + y); x, y ∈ N}, then f is a function
from N to N.
Which of the statements given above is/are correct?
(a)
(b) II only
I only
(c)
Both I and II
(d)
Neither I nor II
Question 59 : Which of the arguments is valid in proving sum of
“two odd numbers is not odd”.
(a)
5 + 5 = 10, hence true for all.
(b)
(c)
All of the mentioned.
(d)
(2n +1) + (2m +1) = 2(n+m+1)
hence true for all.
None of the mentioned.
Question 60 : Let A = {1, 2, 3, 4} and R be given by the digraph shown below.
Which of the following relations determined by this digraph:
(a)
(c)
{(1, 1), (1, 3), (1,4), (2, 1), (2, 3), (2, 4), (3, 1),
(4, 1)}
{(1, 1), (1, 3), (2, 1), (2, 3), (2, 4), (3, 1), (3, 2),
(4, 1)}
(b)
{(1, 1), (1, 2), (1,3), (1,4), (2, 1), (2, 3),
(2, 4), (3, 1), , (3, 2)}
(d)
None of the mentioned.
Question 61: Let A = {a, b, c, d, e, f , g, h} Consider the following subsets of A
A1 = {a, c, e, f, g, h}, A2 = {a, c, e, g}, A3 = {b, d}, A4 = { f, h} and A5=Φ. Then find the
partition of A.
(a)
(c)
A1 , A3 and A4
A2 , A3 and A4
(b)
(d)
A1 , A3 and A5
None of the above.
Question 62 : Let A = {1, 2, 3} and R={(1, 2), (2, 3)} be a relation in A. Then,
the minimum number of ordered pairs may be added, so that R becomes an
equivalence relation, is
(a)
1
(b)
4
(c)
5
(d)
7
10
Question 63 : Draw a graph with the adjacency matrix
0
1

1

1
1 1 1
0 1 0
1 0 0

0 0 0
with respect to the ordering of vertices a, b, c, d.
(a)
(b)
(c)
(d)
None of the above
Question 64: Let A be a finite set and R  AA be a relation on A (Domain R=A), then
Match the following:
A
If x, y, z[((x, y)  R  (x, z)  R)  y  z] , then
1.
R is called transitive.
B
If x, y[((x, y)  R  (y, x)  R)  x  y] , then
2.
R is not symmetric
C
If x, y,z[((x, y)R  (y,z)R)  (x,z)R] , then
3.
R : A  A
D
If xy[(x, y)  R  (y, x)  R] , then
4.
R is called antisymmetric
Codes: ABCD
(a)
3421
(b)
3241
(d)
(c)
3412
4312
Question 65 : “There is one and only one x such that:P(x)” can be expressed as:
(a)
x (P(x) ∧ y (P(y)  y = x))
(b)
x (P(x)  y (P(y)  y = x))
(c)
x (P(x) ∧y (P(y)  y = x))
(d)
x (P(x) ∧ y (P(y) ∧ y ≠ x))
Question 66 : Let A, B and R be three propositions, then which of the following is a
Tautology?
(a)
[(A  B)  r]  [A  (B  R)]
(b)
[(A ∧ B)  R]  [(A  R) ∧ (B  R)]
(c)
[(A ∨ B) ∧ (A  R) ∧ (B  R)]  R
(d)
None of the above.
11
Question 67 : Which of the following sequences is not graphic.
(a)
0, 0, 0, 0, 0, 0
(b)
4, 4, 4, 4, 4, 0
(c)
5, 3, 3, 3, 3, 3
(d)
5, 5, 4, 3, 2, 1
Question 68 : Let us consider the propositional function B(x,y). The negation of:
∀x∃y[B(x, y) ∧ ∀z((z ≠ y) →  B(x, z))] is…..
∃x∀y [B(x, y)  ∃z(z≠y ∧ B(x,z)]
(b) ∃x∀y [B(x, y)  ∃z(z=y ∧ B(x,z)]
(d) non of the above.
∃x∀y [B(x, y)  ∃z(z≠y ∧ B(x,z)]
(a)
(c)
Question 69 : Let the relations R and S be represented by the matrices
1
M R   1
 0
0
0
1
1
0 
0 
and
MS
1
  0
 1
0
1
0
1
1 
0 
What are the matrix representing RS?
(a)
1
1

1
(c)
1
1

1
0
1
0
0
1
1
1
1 
0 
1
1 
0 
(b)
1
0

 0
(d)
None of the above.
0
0
0
1
0 
0 
Question 70 : The size of a Simple Graph is the …….
(a)
(c)
cardinality of edges
sum of degree
(b)
(d)
Cardinality of vertices
degree sequence.
Question 71 : Let S(K) = 1 + 3 + 5 + ... + (2K - 1) = 3 + K2, then which of the following is
true?
(a)
Principle of mathematical induction can
be used to prove the formula
(b)
S(K)  S(K + 1)
(c)
S(K) ∧  S(K + 1)
(d)
S(1) is correct.
Question 72 : A function f: Z → N , where N = {0,1,2,….} defined by f(x) = x2 is ……….
(a)
(c)
injective but not surjective
bijective
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(b)
surjective but not injective
(d)
None of the above.
Question 73 : Determine if the following argument is valid or Invalid.
If the program crashed, an exception was raised. If an exception was raised, someone
input a text value for an integer value. Therefore, the program did not crash,
(a)
Question 74:
(b)
Invalid
Valid
Use incidence matrix to represent the graph below.
e5
a
e6
e8
b
e2
d
e7
e4
e1
c
e3
with respect to the ordering of vertices a, b, c, d.
-1
0

1

0
1
0

-1

0
(a)
(c)
0 0 0
1 0 1
0 1 -1
-1 -1 0
0 0 0
-1 0 -1
0 -1 1
1 1 0
1 1 0 0
-1 0 1 0 
0 0 0 0

0 -1 0 1
- 1 -1 0 0 
1 0 1 0 
0 0 0 0

0 1 0  1
(b)
1
0

-1

0
0 0 0 -1 -1 0 0
-1 0 -1 1 0 1 0
0 -1 1 0 0 0 0

1 1 0 0 1 0 1
(d)
None of the above
Question 75 : Let R be a binary relations on N x N. Decide which of the given ordered
pair belongs to R.
A xRy iff
B xRy iff
C xRy iff
D xRy iff
Codes: ABCD
(a)
4123
(c)
By Match the following:
1.
(6,21)
x = y+1;
2.
(2,28)
x ≡3 y;
2
3.
(4,2)
x=y ;
4.
(5,4)
x divides y;
(b)
4231
(d)
4213
4132
Question 76 : Let p denote the statement: “Amr is tall”, q: “Amr is handsome”. Then
the statement “Amr is tall, but not handsome”, in symbolic form is …..
(a) ¬ q ˄ p
(b) ¬ p ∨ q
(c)
¬p∨ ¬q
(d) ¬ p ˄ ¬ q
13
Question 77 : Let p and q be two propositions, then which of the following are
logically equivalent?
(a)
(p → q)  r and p  (q  r)
(b)
(p ∧ q)  r and (p  r) ∧ (q  r)
(c)
(p  q) ∨ (p  r) and p  (q ∨ r)
(d)
None of the above.
Question 78 : If A = {1, 2, 3, 4, 5, 7, 8, 9} and B = {2, 6} then find the number of
proper subsets of A ∩ B ?
(a)
1
(b)
2
(c)
15
(d)
31
Question 79: Let us consider the propositional function P(x). The negation of:
∃xP(x) ∧ ∀x∀y( P(x) ∧ P(y)  x=y) is…..
(a) ∀x(P(x))  ∃x∃y [P(x) ∧ P(y) ∧ x=y] (b) ∀x(P(x))  ∃x∃y [P(x)  P(y)  x≠y]
(c) ∀x(P(x))  ∃x∃y [P(x) ∧ P(y) ∧ x≠y] (d) non of the above.
Question 80 : Let A = {1, 2, 3} and R={(1, 2), (2, 3)} be a relation in A. Then, the
minimum number of ordered pairs may be added, so that R becomes an equivalence
relation, is…….
(a)
(b) 6
5
(c)
(d) 8
7
Question 81 : Which of the directed graphs representing the relations
R = {(1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 3), (4, 1), (4, 3)}.
(a)
(b)
(c)
(d)
None of the mentioned.
Question 82 : Let A and B be any sets. Which of the following statements is True ?
(a)
(c)
  P ( 2,3  ) and   P ( 2,3  ) ,
where P ( 2,3  ) is the power set of
 2 A
 2,3 .
 3 A
implies that  2 , 3   A .
and
(b)
(d)
14
2  A
 3   A B
implies that  2    3   A  B .
and
None of the above
Question 83 : Evaluate the following proof that, for any sets A, B : (A  B) C
 AC  BC
Proof: Take x  (A  B) C . Hence x  A  B . So x  A and x  B .
Therefore, x  A C and x  B C . Hence x  A C  B C .
(a)
(c)
The theorem and proof are
correct.
The proof is correct, but the
theorem is false.
(b)
(d)
The theorem is correct, but the
proof is in error.
The theorem is false, and the proof
is in error.
Question 84 : Determine if the argument is valid.
1-
If a number is even, then it is divisible by 2.
23-
If a number is divisible by 8, then it is divisible by 2.
Some numbers are either even or divisible by 8.
Therefore : ∃x(x is a number divisible by 2).
(a)
(b)
Invalid
Valid
Question 85 : The adjacency matrix of the following graph.
with respect to the ordering of vertices a, b, c, d.
(a)
(c)
0
1

1

1
0
1

1

1
1 1 1
0 0 1 
0 0 1

1 1 0
1 1 1
0 0 1 
0 0 1

0 1 0
15
(b)
0
1

1

1
1 1 1
0 1 0
1 0 0

0 0 0
(d)
None of the above
Question 86 : Let the universe of discourse be all subsets of the real numbers R .
If P(A,B) denotes “ A  B   ”. Match the following:
A
B
C
D
BA P(A,B)
AB P(A,B)
AB P(A,B)
BA P(x,y)
Codes: ABCD
(a)
1121
(c)
1.
True
2.
False
(b)
1222
(d)
1112
None of the above
Question 87 : Let P: buy a car from GMC Motor, Q: get $2000 cash back and
R: get a 2% car loan, then the statement form:
“When you buy a new car from GMC Motor, you get $2000 back in cash or a 2% car
loan“ , in symbolic from is …..
(a)
P QR
(b)
PQ∧R
(c)
P Q R
(d)
P ∧ (Q  R)
Question 88 : Let us consider the propositional functions P(x) and Q(x,y). The
(a)
(c)
negation of:
∀ x [ P(x)  ∃ y ( P(y) ∧ Q(x,y)) ]
is…..
∃x [∀y (Q(x,y)   P(y)) ∧ P(x)]
(b) ∃x[P(x)  ∀y (P(y) ∧Q(x,y))]
∃ x [ P(x)  ∃ y ( P(y) ∧ Q(x,y)) ] (d) non of the above.
Question 89 :
If
P Q
P R
 R
‫ــــــــــــــــــــ‬
Therefore ………….
(a)
Q
(b)
P
(c)
S
(d)
None of the above.
16
Question 90 : Let the relations R and S be represented by the matrices
1
M R   1
 0
0
0
1
1
0 
0 
and
MS
1
  0
 1
0
1
0
1
1 
0 
What are the matrix representing RS?
(a)
(c)
1
1

1
1
1 1 
1 0 
1 0 1 
0 0 0 


 0 0 1 
(b)
1
0

 0
(d)
None of the above.
0
0
0
0
1
0 
0 
Question 91 : Determine the set A   x  Z : x 7 3   26  x  26 
(a)
(c)
{ −19, −12, −5, 2, 9, 16, 23}
{ -21, -14, -7, 0, 7, 14, 21}
(b)
(d)
{−25, −18, −11, −4 , 3, 10, 17, 24}
{−20, −13, −6, 1, 8, 15, 22}
Question 92 : Let X be any set, R be a relation in P(X), defined by :
“ (A, B)  R iff A  B ”. Then, R is
(a)
(c)
A function from P(X) to P(X).
(b)
symmetric and transitive.
(d)
reflexive and symmetric.
transitive and antisymmetric.
Question 93 : Let A and B be any sets. Which of the following statements is True ?
(a)
(c)
If   A , then   P (A ) ,
(b)
where P (A) is the power set of A .
 2 A
and
 3 A
implies that
(d)
 2, 3   A.
2  A
 3   BA
implies that  2    3   A  B .
and
None of the above
Question 94 : A function f: Z → Z defined by f(x) = x2 is ……….
(a)
(c)
injective but not surjective
bijective
(b)
surjective but not injective
(d)
None of the above.
Question 95 : Which of the following sequences is graphic.
(a)
6, 5, 4, 3, 2, 1
(b)
5, 5, 4, 3, 2, 1
(c)
1, 1, 1, 1, 1, 1
(d)
None of the above.
17
Question 96 : Draw a graph with the incidence matrix
-1
0

1

0
1 1 0 0
-1 0 1 0 
0 1 -1 0 0 0 0 

-1 -1 0 0 -1 0 1
0 0 0
1 0 1
with respect to the ordering of vertices a, b, c, d.
(a)
(b)
(c)
(d) None of the above
Question 97 : Let us consider the propositional function B(x,y). The negation of:
(a)
(c)
∀x∃y[B(x, y) ∧ ∀z((z ≠ y) →  B(x, z))] is…..
∃x∀y [B(x, z)  ∃z(z≠y ∧ B(x,z)]
(b) ∃x∀y [B(x, z)  ∃z(z=y ∧ B(x,z)]
(d) non of the above.
∃x∀y [B(x, z)  ∃z(z≠y ∧ B(x,z)]
Question 98 : Let the universe of discourse be all people, S(x) denoting “x is a
student in FCAI”, and define a propositional function J(x) denoting “x has taken a
course in Java”. Translate the following sentence into predicate logic:
“Every student in FCAI has taken a course in Java.”
(a)
(b) x(S(x) ∧ J(x))
xJ(x)
(c)
(d) none of the above.
x (S(x)→ J(x))
Question 99 :
Fact 1: Ahmed said, "Ann and I both have cats."
Fact 2: Amr said, "I don't have a cat."
Fact 3: Ahmed always tells the truth, but Amr sometimes lies.
If the first three statements are facts, which of the following statements must also be a
fact?
I: Amr has a cat.
II: Ahmed has a cat.
III: Amr is lying.
(a)
All the statements are facts.
(b)
I only
(c)
II only
(d)
I and II only
18