In the name of ALLAH, the most beneficient, the most merciful

## Discrete Mathematics (MTH202)

Multiple Choice Questions (MCQs)

## Objective Questions

### Question

A Random variable is also called a _________.

1. Chance Variable
2. Constant

### Question

If X and Y are independent random variables, then E(XY) is equal to

1. E(XY)
2. XE(Y)
3. YE(X)
4. E(x)E(y)

### Question

If X and Y are random variables, then E(aX) is equal to

1. E(aX)
2. aE(X)
3. aX
4. None of the given

### Question

If X and Y are independent random variables and a and b are constants, then Var(aX + bY)is equal to

1. aVar(X) + bVar(Y)
2. (a + b)[Var(X) + Var(Y)]
3. Var(aX) + Var(bY)
4. a^2 Var(X) + b^2 Var(Y)

### Question

If A and B be events with P(A) = 1/3, P(B) = 1/4 and P(A ∩ B) = 1/6, then P(A ∪ B) = ________ .

1. 2/3
2. 5/12
3. 1/24
4. 1/2

### Question

What is the minimum number of students in a class to be sure that two of them are born in the same month?

1. 11
2. 12
3. 13
4. 14

### Question

Let A and B be subsets of U with n(A) = 12, n(B) = 15, n(A') = 17, and n(A intersection B) = 8, then n(U)=______ .

1. 27
2. 29
3. 20
4. 35

### Question

Which of the followings is the product set A * B * C ? where A = {a}, B = {b}, and C = {c, d}.

1. {(a, b, c), (a, b, d)}
2. {(a, c, b), (a, d, b)}
3. {(b, c, a), (b, d, a)}
4. {(c, b, a), (d, b, a)}

### Question

If A and B are disjoint finite sets then n(A ∪ B) = ______.

1. n(A) − n(B)
2. n(A) + n(B) − n(A ∩ B)
3. n(A) + n(B)
4. n(A) + n(B) + n(A ∩ B)

### Question

Among 20 people, 15 either swim or jog or both. If 5 swim and 6 swim and jog, how many jog?

1. 6
2. 16
3. 24
4. 46

### Question

A tree is normally constructed from ________.

1. right
2. center
3. left to right
4. right to left

### Question

Find the number of distinct permutations that can be formed using the letters of the word ”BENZENE”

1. 120
2. 220
3. 320
4. 420

### Question

The number of the words that can be formed from the letters of the word,“COMMITTEE” are

1. 9p9
2. 9C9
3. 9! / (2!2!2!)
4. None of the given

### Question

Find the number of the word that can be formed of the letters of the word “ELEVEN”.

1. 120
2. 110
3. 220
4. None of the given

### Question

A student is to answer five out of nine questions on exams. Find the number of ways that can choose the five questions.

1. 216
2. 316
3. 126
4. None of the given

### Question

Let X = {1, 2, 3}, then 2-combinations of the 3 elements of the set X are _________?

1. {1, 2}, {1, 3} and {2, 3}
2. {1, 2}, {2, 1}, {1, 3}, {3, 1}, {2, 3}, and {3, 2}
3. {1, 2}, {2, 1}, {1, 3} and {2, 3}
4. {1, 2}, {2, 1},{1, 3} and {3, 1}

### Question

(-2)! = _________ ?

1. -2
2. 0
3. 2
4. Undefined

### Question

How many possible outcomes are there when a fair coin is tossed four times?

1. 4
2. 8
3. 16
4. 32

### Question

A box contains 5 different colored light bulbs. Which of the followings is the number of ordered samples of size 3 with replacement?

1. 8
2. 15
3. 125
4. 243

### Question

In how many ways can 6 people be seated on 6 available seats?

1. 120
2. 6
3. 12
4. 720

P(0, 0)=______?

1. 0
2. 1
3. 2
4. undefined

### Question

If order matters and repetition is allowed, then which counting method should be used in order to select 'k' elements from a total of 'n' elements?

1. K-Selection
2. K-Sample
3. K-combination
4. K-Permuatation

### Question

In how many ways a student can choose a course from 2 science courses,3 literature courses and 5 art courses.

1. 30
2. 10
3. 1440
4. 240

### Question

Which of the followings is the factorial form of 5 . 4?

1. 5/3
2. 5!/3
3. 5!/3!
4. 5/3!

### Question

In how many ways a student can choose one of each of the courses when he is offered 3 mathematics courses, 4 literature courses and 2 history courses.

1. 9
2. 24
3. 288
4. 14

### Question

Suppose there are 8 different tea flavors and 5 different biscuit brands. A guest wants to take one tea and one brand of biscuit. How many choices are there for this guest?

1. 5
2. 8
3. 13
4. 40

### Question

There are 5 girls students and 20 boys students in a class. How many students are there in total ?

1. 4
2. 15
3. 25
4. 100

### Question

A student can choose a computer project from one of the two lists. The two lists contain 12 and 18 possible projects, respectively. How many possible projects are there to choose from?

1. 12
2. 18
3. 30
4. 216

### Question

There are three bus lines between A and B, and two bus lines between B and C. Find the number of ways a person can travel round trip by bus from A to C by way of B?

1. 5
2. 6
3. 10
4. 36

### Question

A non-zero integer d divides an integer n if and only if there exists an integer k such that _________.

1. n = d / k
2. n = d k
3. n = d + k
4. n = d - k

### Question

Which of the following statements is true according to the Division Algorithm?

1. 17 = 5 x 1 + 12
2. 17 = 5 x 3 + 2
3. 17 = 5 x 4 - 3
4. 17 = 5 x 5 - 8

### Question

A predicate becomes _________ when its variables are given specific values.

1. sentence
2. statement
3. algorithm
4. iteration

### Question

The method of loop invariants is used to prove __________ of a loop with respect to certain pre and post-conditions.

1. falseness
2. correctness

### Question

The set of prime numbers is _________.

1. finite set
2. infinite set
3. continuous set
4. None of the given

### Question

Reductio and absurdum' is another name of _________.

1. Direct Method of proof
3. proof by contapositive
4. None of the given

### Question

If r is a positive real number, then the value of r in 3.r.r = −27r is ________.

1. −9
2. +9
3. 0
4. None of the given

### Question

An integer n is a perfect square if and only if ________ for some integer k.

1. n = 2k
2. n = k^2
3. n = square-root of k
4. n = k^3

### Question

The total number of terms in an arithmetic series 0 + 5 + 10 + 15 + .... + 50 are ________.

1. 9
2. 10
3. 11
4. infinite

### Question

Let f(x)=3x and g(x) = 3x − 2 define functions f and g from R to R. Then (f+g)(x) = ________.

1. −2
2. 6x + 2
3. 6x − 2
4. 6x.x − 2

### Question

Real valued function is a function that assigns _______ to each member of its domain.

1. negative real number
2. positive real number
3. only a real number
4. any arbitrary real number

### Question

A set is called finite, if and only if, it is the ________ or there is ________ .

1. empty set, onto
2. empty set, one-to-one
3. one-to-one, onto
4. empty set, bijective

### Question

If f and g are two one-to-one functions, then their composition that is gof is one-to-one.

1. TRUE
2. FALSE

### Question

Let f(x) = x2 + 1 define functions f from R to R and c = 2 be any scalar, then c.f(x) is ______.

1. 2
2. x2 + 1
3. 2x2 - 1
4. 2x2 + 2

### Question

The set Z of all integers is _____.

1. uncountable
2. countable

### Question

Let f(x) = 3x and g(x) = x + 2 define functions f and g from R to R, then (f.g)(x) is _____.

1. 2x − 2
2. 3x + 2
3. 4x + 2
4. 3x2 + 6x

### Question

If a function (g o f)(x):X→Z is defined as (g o f)(x) = g(f(x)) for all x ∈ X. Then the function ________ is known as composition of f and g.

1. (f o g)
2. f-1(g(x))
3. (g o f)
4. g-1(f(x))

### Question

Let g be a function defined by g(x) = x + 1. Then the composition of (g o g)(x)is ______.

1. x
2. x + 1
3. x + 2
4. x2 + 2x + 1

### Question

The functions 'f' and 'g' are inverse of each other if and only if their composition gives _______.

1. constant function
2. identity function
3. bijective function
4. injective function

### Question

The functions f o g and g o f are always equal.

1. TRUE
2. FALSE

### Question

One-to-One correspondence means the condition of ______.

1. one-One
2. identity
3. onto
4. one-One and onto

### Question

Let A = {1, 2, 3, 4} and B = {7} then the constant function from A to B is _________ .

1. Onto
2. One to one
3. Both one to one and onto
4. Neither one to one nor onto

### Question

Let A = {1, 2, 3} and B = {2, 4} then number of functions from A to B are _________.

1. 6
2. 8
3. 16
4. 64

### Question

For the following relation to be a function, x can not be what values?
R = {(2,4), (x,1), (4,2), (5,6)}

1. x cannot be 2, 4 or 5
2. x cannot be 4, 1 or 6
3. x cannot be 2, 4 or 6
4. x cannot be 1, 2 or 6

### Question

Let X = {2, 4, 5} and Y= {1, 2, 4} and R be a relation from X to Y defined by R = {(2,4), (4,1), (a,2)}. For what value of ‘a‘ the relation R is a function ?

1. 1
2. 2
3. 4
4. 5

### Question

Which relations below are functions?
R1 = {(3,4), (4,5), (6,7), (8,9)}
R2 = {(3,4), (4,5), (6,7), (3,9)}
R3 = {(-3,4), (4,-5), (0,0), (8,9)}
R4 = {(8,11), (34,5), (6,17), (8,19)}

1. R1 and R3 are functions
2. R1 and R2 are functions
3. R2 and R4 are functions
4. R3 and R2 are functions

### Question

Let R be a relation on a set A. If R is reflexive then its compliment is ________ .

1. Reflexive
2. Irreflexive
3. Symmetric
4. Antisymmetric

### Question

R = {(a,1), (b,2), (c,3), (d,4)} then the inverse of this relation is _______.

1. {(a,1), (b,2), (3,c), (4,d)}
2. {(1,a), (2,b), (3,c), (4,d)}
3. {(a,1), (2,b), (3,c), (4,d)}
4. {(1,a), (b,2), (3,c), (4,d)}

### Question

Let R be a relation on a set A. If R is symmetric then its compliment is __________.

1. Reflexive
2. Irreflexive
3. Symmetric
4. Antisymmetric

### Question

Let R be a relation on a set A. If R is reflexive then its compliment is ____________.

1. Reflexive
2. Irreflexive
3. Symmetric
4. Antisymmetric

### Question

Let R be the universal relation on a set A then which one of the following statement about R is true?

1. R is not symmetric
2. R is not reflexive
3. R is not transitive
4. R is reflexive, symmetric and transitive.

### Question

Range of the relation {(0,1), (3,22), (90,34)} is __________ .

1. {0, 3, 90}
2. {1, 22, 34}
3. {0, 1, 3}
4. {0, 1, 3, 22, 90, 34}

### Question

Which of the followings is the product set A * B * C ? where A = {a}, B = {b}, and C = {c, d}.

1. {(a, b, c), (a, b, d)}
2. {(a, c, b), (a, d, b)}
3. {(b, c, a), (b, d, a)}
4. {(c, b, a), (d, b, a)}

### Question

Determine values of x and y, where (2x, x + y) = (8, 6).

1. x = 3 and y = 5
2. x = 4 and y = 2
3. x = 6 and y = 12
4. x = 4 and y = 12

### Question

x belongs to A or x belongs to B, therefore x belongs to ________.

1. A intersection B
2. A union B
3. A difference B
4. A symmetric difference B

### Question

If A and B are any two sets, then A − B = B − A

1. True
2. False

### Question

Let A = {2, 3, 5, 7}, B = {2, 3, 5, 7, 2}, C = Set of first five prime numbers. Then from the following which statement is true ?

1. A = B
2. A = C
3. B = C
4. All the three sets are equal.

### Question

If A = Set of students of virtual university then A has been written in the _________.

1. Tabular form
2. Set builder form
3. Descriptive form
4. A is not a set

### Question

The switches in parallel act just like ________.

1. NOT gate
2. AND gate
3. OR gate
4. XOR gate

### Question

Let p1, p2, p3 be True premises in a given Truth Table. If the conjunctions of the Conclusion with each of p1, p2, p3 are True, then the argument is ________.

1. False
2. True
3. Invalid
4. Valid

### Question

'p is equivalent to q' means ________.

1. p is not necessary but p is sufficient for q.
2. p is neither necessary nor sufficient for q.
3. p is necessary and sufficient for q.
4. p is necessary but not sufficient for q.

### Question

What is the truth value of the sentence?
'It rains if and only if there are clouds.'

1. True
2. False

### Question

If p is false and q is true, then ∼p ↔ q is ________.

1. True
2. False

### Question

If p ↔ q is True, then ________.

1. Only p is True.
2. Only q is True.
3. p and q both are True.
4. None of the given.

### Question

If p is false and q is false, then ∼p implies q is ________.

1. True
2. False

### Question

The converse of the conditional statement 'If I live in Quetta, then I live in Pakistan' is ________.

1. If I live in Pakistan, then I live in Quetta.
2. If I live in Pakistan, then I do Not live in Quetta.
3. If I do Not live in Quetta, then I do Not live in Pakistan
4. If I do Not live in Quetta, then I live in Pakistan

### Question

∼(P → q) is logically equivalent to _________.

1. p ∧ ∼q
2. p ∨ ∼q
3. ∼p ∧ q
4. ∼p ∨ q

### Question

Let p be True and q be True, then ( ∼p ∧ q ) is ________.

1. t ( where t is tautology. )
2. c ( where c is contradiction. )
3. True
4. False

### Question

The contrapositive of the conditional statement 'If it is Sunday, then I go for shopping' is ________.

1. I do Not go for shopping, then it is Not Sunday.
2. I go for shopping, then it is Sunday.
3. I do Not go for shopping, then it is Sunday.
4. I go for shopping, then it is Not Sunday.

### Question

The converse of the conditional statement p → q is

1. q → p
2. ∼q → ∼p
3. ∼p → ∼q
4. None of the given

### Question

The statement p → q is logically equivalent to ∼q → ∼p

1. True
2. False

### Question

Let p → q be a conditional statement, then the statement q → p is called ________.

1. Inverse
2. Converse
3. Contrapositive
4. Double conditional

### Question

( p ∨ ∼p ) is the ________.

2. Conjunction
3. Tautology
4. Contingency

### Question

If p = It is raining, q = She will go to college
"It is raining and she will not go to college”
will be denoted by

1. p ∧ ∼q
2. p ∧ q
3. ∼(p ∧ q)
4. ∼p ∧ q

### Question

The negation of “Today is Friday” is

1. Today is Saturday
2. Today is not Friday
3. Today is Thursday
4. None of the given

### Question

The disjunction of p and q is written as ________.

1. p ∨ q
2. p ∧ q
3. p XOR q
4. None of the given

### Question

The logical statement p ∧ q means ________.

1. p OR q
2. p NOT q
3. p AND q
4. p XOR q

### Question

The disjunction p ∨ q is False when ________.

1. p is False, q is True.
2. p is True, q is False.
3. p is True, q is True.
4. p is False, q is False.

### Question

The conjunction p ∧ q is True when _________.

1. p is True, q is False
2. p is False, q is True
3. p is True, q is True
4. p is False, q is False