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## Discrete Mathematics MCQ Set 1

1. For the sequence 1, 7, 25, 79, 241, 727 … simple formula for {an} is ____________
a) 3n+1 – 2
b) 3n – 2
c) (-3)n + 4
d) n2 – 2

Answer: b [Reason:] The ratio of consecutive numbers is close to 3. Comparing these terms with the sequence of {3n} which is 3, 9, 27 …. Comparing these terms with the corresponding terms of sequence {3n} and the nth term is 2 less than the corresponding power of 3.

2. For the sequence 0, 1, 2, 3 an is ____________
a) ⌈n/2⌉+⌊n/2⌋
b) ⌈n/2⌉+⌈n/2⌉
c) ⌊n/2⌋+⌊n/2⌋
d) ⌊n/2⌋

Answer: a [Reason:] Expand the sequence ⌈n/2⌉+⌊n/2⌋ where a1 is ⌊0.5⌋+⌈0.5⌉ = 1+0 = 1, a2 is ⌊1⌋+⌈1⌉ = 1 + 1 = 2 and so on.

3. The value of∑(k=50)100 k2 is __________
a) 338,350
b) 297,900
c) 297,925
d) 290,025

Answer: c [Reason:] Using the formula .∑(k=1)n k2 = (n(n + 1)(2n + 1)) / 6.

4. The sets A and B have same cardinality if and only if there is ___________ from A to B.
a) One-to-one
b) One-to-many
c) Many-to-many
d) Many-to-one

Answer: a [Reason:] If there is one-to-one correspondence then they have same cardinality.

5. For the sequence an = ⌊√2n+ 1/2⌋, a7is ____________
a) 1
b) 7
c) 5
d) 4

Answer: d [Reason:] a7 = ⌊√14+1/2⌋ which is ⌊4.24⌋ = 4.

6. The value of ∑(i=1)3 ∑(h=0)2 i is _________
a) 10
b) 17
c) 15
d) 18

Answer: d [Reason:] The value of ∑(i=1)3 ∑(h=0)2 i = 1+1+1+2+2+2+3+3+3 = 18.

7. For the sequence an = 6. (1/3)n, a4 is _________
a) 2/25
b) 2/27
c) 2/19
d) 2/13

Answer: b [Reason:] Put n = 4 in the sequence.

8. The value of ∑(i=0)4i! is __________
a) 32
b) 30
c) 34
d) 35

Answer: c [Reason:] First five term of the sequence n! is given by 1, 1, 2, 6, 24.

9. Set of all integers is counter. Is it True or False?
a) True
b) False

Answer: a [Reason:] There is one-to-one correspondence between set of positive integers and set of all integers.

10. The value of ∏( k=1)100 (-1) k is _________
a) 0
b) 1
c) -1
d) 2

Answer: b [Reason:] The product of a1, a2, a3 …… an is represented by ∏(i=1)n ai.

## Discrete Mathematics MCQ Set 2

1. The union of the sets {1, 2, 5} and {1, 2, 6} is the set _______________
a) {1, 2, 6, 1}
b) {1, 2, 5, 6}
c) {1, 2, 1, 2}
d) {1, 5, 6, 3}

Answer: b [Reason:] The union of the sets A and B, is the set that contains those elements that are either in A or in B.

2. The intersection of the sets {1, 2, 5} and {1, 2, 6} is the set _____________
a) {1, 2}
b) {5, 6}
c) {2, 5}
d) {1, 6}

Answer: a [Reason:] The intersection of the sets A and B, is the set containing those elements that are in both A and B.

3. Two sets are called disjoint if there _____________ is the empty set.
a) Union
b) Difference
c) Intersection
d) Complement

Answer: c [Reason:] By the definition of the disjoint set.

4. Which of the following two sets are disjoint?
a) {1, 3, 5} and {1, 3, 6}
b) {1, 2, 3} and {1, 2, 3}
c) {1, 3, 5} and {2, 3, 4}
d) {1, 3, 5} and {2, 4, 6}

Answer: d [Reason:] Two sets are disjoint if the intersection of two sets is the empty set.

5. The difference of {1, 2, 3} and {1, 2, 5} is the set ____________
a) {1}
b) {5}
c) {3}
d) {2}

Answer: c [Reason:] The difference of the sets A and B denoted by A-B, is the set containing those elements that are in A not in B.

6. The complement of the set A is _____________
a) A – B
b) U – A
c) A – U
d) B – A

Answer: b [Reason:] The complement of the set A is the complement of A with respect to U.

7. The bit string for the set {2, 4, 6, 8, 10} (with universal set of natural numbers less than or equal to 10) is ____________________
a) 0101010101
b) 1010101010
c) 1010010101
d) 0010010101

Answer: a [Reason:] The bit string for the set has a one bit in second, fourth, sixth, eighth, tenth positions, and a zero elsewhere.

8. Let Ai = {i, i+1, i+2, …..}. Then set {n, n+1, n+2, n+3, …..} is the _________ of the set Ai.
a) Union
b) Intersection
c) Set Difference
d) Disjoint

Answer: b [Reason:] By the definition of the generalized intersection of the set.

9. The bit strings for the sets are 1111100000 and 1010101010. The union of these sets is ___________
a) 1010100000
b) 1010101101
c) 1111111100
d) 1111101010

Answer: d [Reason:] The bit string for the union is the bitwise OR of the bit strings.

10. The set difference of the set A with null set is __________
a) A
b) null
c) U
d) B

Answer: a [Reason:] The set difference of the set A by null set denoted by A – {null} is A.

## Discrete Mathematics MCQ Set 3

1. A __________ is an ordered collection of objects.
a) Relation
b) Function
c) Set
d) Proposition

Answer: c [Reason:] By the definition of set.

2. The set O of odd positive integers less than 10 can be expressed by _____________
a) {1, 2, 3}
b) {1, 3, 5, 7, 9}
c) {1, 2, 5, 9}
d) {1, 5, 7, 9, 11}

Answer: b [Reason:] Odd numbers less than 10 is {1, 3, 5, 7, 9}.

3. Power set of empty set has exactly _________ subset.
a) One
b) Two
c) Zero
d) Three

Answer: a [Reason:] Power set of null set has exactly one subset which is empty set.

4. What is the Cartesian product of A = {1, 2} and B = {a, b}?
a) {(1, a), (1, b), (2, a), (b, b)}
b) {(1, 1), (2, 2), (a, a), (b, b)}
c) {(1, a), (2, a), (1, b), (2, b)}
d) {(1, 1), (a, a), (2, a), (1, b)}

Answer: c [Reason:] A subset R of the Cartesian product A x B is a relation from the set A to the set B.

5. The Cartesian Product B x A is equal to the Cartesian product A x B. Is it True or False?
a) True
b) False

Answer: b [Reason:] Let A = {1, 2} and B = {a, b}. The Cartesian product A x B = {(1, a), (1, b), (2, a), (2, b)} and the Cartesian product B x A = {(a, 1), (a, 2), (b, 1), (b, 2)}. This is not equal to A x B.

6. What is the cardinality of the set of odd positive integers less than 10?
a) 10
b) 5
c) 3
d) 20

Answer: b [Reason:] Set S of odd positive an odd integer less than 10 is {1, 3, 5, 7, 9}. Then, Cardinality of set S = |S| which is 5.

7. Which of the following two sets are equal?
a) A = {1, 2} and B = {1}
b) A = {1, 2} and B = {1, 2, 3}
c) A = {1, 2, 3} and B = {2, 1, 3}
d) A = {1, 2, 4} and B = {1, 2, 3}

Answer: c [Reason:] Two set are equal if and only if they have the same elements.

8. The set of positive integers is _____________
a) Infinite
b) Finite
c) Subset
d) Empty

Answer: a [Reason:] The set of positive integers is not finite.

9. What is the Cardinality of the Power set of the set {0, 1, 2}.
a) 8
b) 6
c) 7
d) 9

Answer: a [Reason:] Power set P ({0, 1, 2}) is the set of all subsets of {0, 1, 2}. Hence,P({0, 1, 2}) = {null , {0}, {1}, {2}, {0, 1}, {0,2}, {1, 2}, {0, 1, 2}}.

10. The members of the set S = {x | x is the square of an integer and x < 100} is ________________
a) {0, 2, 4, 5, 9, 58, 49, 56, 99, 12}
b) {0, 1, 4, 9, 16, 25, 36, 49, 64, 81}
c) {1, 4, 9, 16, 25, 36, 64, 81, 85, 99}
d) {0, 1, 4, 9, 16, 25, 36, 49, 64, 121}

Answer: b [Reason:] The set S consists of the square of an integer less than 10.

## Discrete Mathematics MCQ Set 4

1. A polygon with 7 sides can be triangulated into
a) 7
b) 14
c) 5
d) 10

Answer: c [Reason:] A simple polygon with n sides can be triangulated into n-2 triangles, where n > 2.

2. Every simple polynomial has an interior diagonal.
a) True
b) False

Answer: a [Reason:] By using Strong Induction.

3. A polygon with 12 sides can be triangulated into
a) 7
b) 10
c) 5
d) 12

Answer: b [Reason:] A simple polygon with n sides can be triangulated into n-2 triangles, where n > 2.

4. Let P(n) be the statement that a postage of n cents can be formed using just 3-cents stamps and 5-cents stamps. Is the statements P(8) and P(10) are Correct?
a) True
b) False

Answer: a [Reason:] We can form 8 cent of postage with one 3-cent stamp and one 5-cent stamp. P(10) is true because we can form it using two 5-cent stamps.

5. Which amount of postage can be formed using just 4-cent and 11-cent stamps?
a) 2
b) 5
c) 30
d) 10

Answer: d [Reason:] We can form 30 cent of postage with two 4-cent stamp and two 11-cent stamp.

6. 22-cent of postage can be produced with two 4-cent stamp and one 11-cent stamp.
a) True
b) False

Answer: b [Reason:] By using two 4-cent stamp and one 11-cent stamp, 27-cent postage is produced.

7. Which amount of postage can be formed using just 3-cent stamp and 10-cent stamps?
a) 27
b) 20
c) 11
d) 5

Answer:a [Reason:] We can form 27 cent of postage with nine 3-cent stamp and 20-cent postage can be formed by using two 10-cent stamps.

8. 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(3)
b) P(2)
c) P(4)
d) P(6)

Answer: a [Reason:] By induction we can prove that P(3) is true but we can’t conclude about P(2), p(6) and P(4).

9. Suppose that P(n) is a propositional function. Determine for which positive integers n the statement P(n) must be true if: P(1) and P(2) is true; for all positive integers n, if P(n) and P(n+1) is true then P(n+2) is true.
a) P(1)
b) P(2)
c) P(4)
d) P(n)

Answer: d [Reason:] By induction we can prove that P(n) is true.

10. A polygon with 25 sides can be triangulated into
a) 23
b) 20
c) 22
d) 21

Answer: a [Reason:] A simple polygon with n sides can be triangulated into n-2 triangles, where n > 2.

## Discrete Mathematics MCQ Set 5

1. Let the statement be “If n is not an odd integer then square of n is not odd.”,then if P(n) is “n is an not an odd integer” and Q(n) is “(square of n) is not odd.” For a direct proof we should proove
a) ∀nP ((n) → Q(n))
b) ∃ nP ((n) → Q(n))
c) ∀n~(P ((n)) → Q(n))
d) ∀nP ((n) → ~(Q(n)))

Answer: a [Reason:] Definition of direct proof.

2. Which of the following can only be used in disproving the statements?
a) Direct proof
b) Contrapositive proofs
c) Counter Example
d) Mathematical Induction

Answer: c [Reason:] Counter examples cannot be used to prove results.

3. Let the statement be “If n is not an odd integer then sum of n with some not odd number will not be odd.”,then if P(n) is “n is an not an odd integer” and Q(n) is “sum of n with some not odd number will not be odd.” A proof by contraposition will be
a) ∀nP ((n) → Q(n))
b) ∃ nP ((n) → Q(n))
c) ∀n~(P ((n)) → Q(n))
d) ∀n(~Q ((n)) → ~(P(n)))

Answer: d [Reason:] Definition of proof by contraposition.

4. When to proof P→Q true, we proof P false, that type of proof is known as
a) Direct proof
b) Contrapositive proofs
c) Vacuous proof
d) Mathematical Induction

Answer: c [Reason:] Definition of vacuous proof.

5. In proving √5 as irrational, we begin with assumption √5 is rational in which type of proof?
a) Direct proof
b) Proof by Contradiction
c) Vacuous proof
d) Mathematical Induction

Answer: b [Reason:] Definition of proof by contradiction.

6. A proof covering all the possible cases, such type of proofs are known as
a) Direct proof
b) Proof by Contradiction
c) Vacuous proof
d) Exhaustive proof

Answer: d [Reason:] Definition of exhaustive proof.

7. Which of the arguments is not valid in proving sum of two odd number is not odd.
a) 3 + 3 = 6 ,hence true for all
b) 2n +1 + 2m +1 = 2(n+m+1) hence true for all
c) All of the mentioned
d) None of the mentioned

Answer: a [Reason:] Some examples are not valid in proving results.

8. A proof broken into distinct cases, where these cases cover all prospects, such proofs are known as
a) Direct proof
b) Contrapositive proofs
c) Vacuous proof
d) Proof by cases

Answer: c [Reason:] Definition of proof by cases.

9. A proof that p → q is true based on the fact that q is true, such proofs are known as
a) Direct proof
b) Contrapositive proofs
c) Trivial proof
d) Proof by cases