250+ TOP MCQs on Inverse of a Function and Answers

Discrete Mathematics Questions and Answers for Entrance exams on “Inverse of a Function”.

1. For an inverse to exist it is necessary that a function should be __________
a) injection
b) bijection
c) surjection
d) none of the mentioned

Answer: b
Clarification: Inverse exist only for those functions which are one one and onto.

2. If f(x) = y then f-1(y) is equal to __________
a) y
b) x
c) x2
d) none of the mentioned

Answer: b
Clarification: On giving inverse, image the function returns preimage thus f-1 (y) = x.

3. A function f(x) is defined from A to B then f -1 is defined __________
a) from A to B
b) from B to A
c) depends on the inverse of function
d) none of the mentioned

Answer: b
Clarification: Inverse associate each element in B with corresponding element in A.

4. If f is a function defined from R to R, is given by f(x) = 3x – 5 then f –1(x) is given by __________
a) 1/(3x-5)
b) (x+5)/3
c) does not exist since it is not a bijection
d) none of the mentioned

Answer: b
Clarification: y = 3x-5, x = (y+5)/3, f -1(x) = (x+5)/3.

5. For some bijective function inverse of that function is not bijective.
a) True
b) False

Answer: b
Clarification: If f(x) is a bijection than f -1(x) is also a bijection.

6. f(x) is a bijection than f -1(x) is a mirror image of f(x) around y = x.
a) True
b) False

Answer: a
Clarification: Inverse of a function is the mirror image of function in line y = x.

7. If f is a function defined from R to R, is given by f(x) = x2 then f -1(x) is given by?
a) 1/(3x-5)
b) (x+5)/3
c) does not exist since it is not a bijection
d) none of the mentioned

Answer: c
Clarification: It is not a one one function hence Inverse does not exist.

8. For any function fof -1(x) is equal to?
a) x
b) 1
c) x2
d) none of the mentioned

Answer: a
Clarification:Compostion of a function with its inverse gives x.

9. The solution to f(x) = f -1(x) are __________
a) no solutions in any case
b) same as solution to f(x) = x
c) infinite number of solution for every case
d) none of the mentioned

Answer: b
Clarification: Inverse of a function is the mirror image of function in line y = x.

10. Let f(x) = x then number of solution to f(x) = f -1(x) is zero.
a) True
b) False

Answer: b
Clarification: Since inverse of a function is the mirror image of function in line y = x, therefore in this case infinte solution will exist.

250+ TOP MCQs on Algorithms – Complexity and Answers

Discrete Mathematics Questions and Answers for Freshers on “Algorithms – Complexity”.

1. Which is used to measure the Time complexity of an algorithm Big O notation?
a) describes limiting behaviour of the function
b) characterises a function based on growth of function
c) upper bound on growth rate of the function
d) all of the mentioned

Answer: d
Clarification: Big O notation describes limiting behaviour, and also gives upper bound on growth rate of a function.

2. If for an algorithm time complexity is given by O(1) then the complexity of it is ____________
a) constant
b) polynomial
c) exponential
d) none of the mentioned

Answer: a
Clarification: The growth rate of that function will be constant.

3. If for an algorithm time complexity is given by O(log2n) then complexity will be ___________
a) constant
b) polynomial
c) exponential
d) none of the mentioned

Answer: d
Clarification: The growth rate of that function will be logarithmic therefore complexity will be logarithmic.

4. If for an algorithm time complexity is given by O(n) then the complexity of it is ___________
a) constant
b) linear
c) exponential
d) none of the mentioned

Answer: b
Clarification: The growth rate of that function will be linear.

5. If for an algorithm time complexity is given by O(n2) then complexity will ___________
a) constant
b) quadratic
c) exponential
d) none of the mentioned

Answer: b
Clarification: The growth rate of that function will be quadratic therefore complexity will be quadratic.

6. If for an algorithm time complexity is given by O((32)n) then complexity will be ___________
a) constant
b) quardratic
c) exponential
d) none of the mentioned

Answer: c
Clarification: The growth rate of that function will be exponential therefore complexity will be exponential.

7. The time complexity of binary search is given by ___________
a) constant
b) quardratic
c) exponential
d) none of the mentioned

Answer: d
Clarification: It is O(log2n), therefore complexity will be logarithmic.

8. The time complexity of the linear search is given by ___________
a) O(log2n)
b) O(1)
c) exponential
d) none of the mentioned

Answer: d
Clarification: It is O(n), therefore complexity will be linear.

9. Which algorithm is better for sorting between bubble sort and quicksort?
a) bubble sort
b) quick sort
c) both are equally good
d) none of the mentioned

Answer: b
Clarification: Running time of quicksort is logarithmic whereas for bubble sort it is quadratic.

10. Time complexity of the binary search algorithm is constant.
a) True
b) False

Answer: b
Clarification: It is O(log2n), therefore complexity will be logarithmic.

250+ TOP MCQs on Principle of Mathematical Induction and Answers

Discrete Mathematics Multiple Choice Questions on “Principle of Mathematical Induction”.

1. What is the base case for the inequality 7n > n3, where n = 3?
a) 652 > 189
b) 42 < 132
c) 343 > 27
d) 42 <= 431

Answer: c
Clarification: By the principle of mathematical induction, we have 73 > 33 ⇒ 343 > 27 as a base case and it is true for n = 3.

2. In the principle of mathematical induction, which of the following steps is mandatory?
a) induction hypothesis
b) inductive reference
c) induction set assumption
d) minimal set representation

Answer: a
Clarification: The hypothesis of Step is a must for mathematical induction that is the statement is true for n = k, where n and k are any natural numbers, which is also called induction assumption or induction hypothesis.

3. For m = 1, 2, …, 4m+2 is a multiple of ________
a) 3
b) 5
c) 6
d) 2

Answer: d
Clarification: For n = 1, 4 * 1 + 2 = 6, which is a multiple of 2. Assume that 4m+2 is true for m=k and so 4k+2 is true based on the assumption. Now, to prove that 4k+2 is also a multiple of 2 ⇒ 4(k+1)+2 ⇒ 2 * 4k – 4k + 6 ⇒ 2*4k+4 – 4k+2 ⇒ 2(4k+2) – 2(2k+1). Here, the first term 2(4k+2) is true as per assumption and the second term 2(4k+2) is must to be a multiple of 2. Hence, 4(k+1)+2 is a multiple of 2. So, by induction hypothesis, (4m+2) is a multiple of 2, for m = 1,2,3,…

4. For any integer m>=3, the series 2+4+6+…+(4m) can be equivalent to ________
a) m2+3
b) m+1
c) mm
d) 3m2+4

Answer: a
Clarification: The required answer is m2+3. Now, by induction assumption, we have to prove 2+4+6+…+4(k+1) = (k+1)2+3 also can be true, 2+4+6+…+4(k+1) = 2+4+6+⋯+(4k+4) and by the subsequent steps, we can prove that (m+1)2+3 also holds for m=k. So, it is proved.

5. For every natural number k, which of the following is true?
a) (mn)k = mknk
b) m*k = n + 1
c) (m+n)k = k + 1
d) mkn = mnk

Answer: a
Clarification: In the first step, for k = 1, (mn)1 = m1n1 = mn, hence it is true. Let us assume the statement is true for k = l, Now by induction assumption, (mn)1 = m1n1 is true. So, to prove, (mn)l+1 = ml + 1nl+1, we have (mn)l = mlnl and multiplying both sides by (mn) ⇒ (mn)1(mn)=(m1n1)(mn)
⇒ (mn)l+1 = (mm1)(nn1) ⇒ (mn)l+1 = (ml+1nl+1). Hence, it is proved. So, (mn)k = mknk is true for every natural number k.

6. By induction hypothesis, the series 12 + 22 + 32 + … + p2 can be proved equivalent to ____________
a) (frac{p^2+2}{7})
b) (frac{p*(p + 1)*(2p + 1)}{6})
c) (frac{p*(p+1)}{4})
d) p+p2

Answer: b
Clarification: By principle of mathematical induction, we now assume that p (b) is true 12 + 22 + 32 + … + b2 = (frac{b (b + 1) (2b + 1)}{6})
so to prove P(b+1): 12 + 22 + 32 + … + b2 + (b + 1)2 = (frac{b (b + 1) (2b + 1)}{6}) + (b + 1)2
By induction assumption it is shown that 12 + 22 + 32 + … + b2 + (b + 1)2 = (frac{(b + 1) [(b + 2) (2b + 3)]}{6}). Hence it is proved that 12 + 22 + 32 + … + p2 = (frac{p*(p + 1)*(2p + 1)}{6}).

7. For any positive integer m ______ is divisible by 4.
a) 5m2 + 2
b) 3m + 1
c) m2 + 3
d) m3 + 3m

Answer: d
Clarification: The required answer is, m3 + 3m. Now, by induction hypothesis, we have to prove for m=k, k3+3k is divisible by 4. So, (k + 1)3 + 3 (k + 1) = k3 + 3 k2 + 6 k + 4
= [k3 + 3 k] + [3 k2 + 3 k + 4] = 4M + (12k2 + 12k) – (8k2 + 8k – 4), both the terms are divisible by 4. Hence (k + 1)3 + 3 (k + 1) is also divisible by 4 and hence it is proved for any integer m.

8. According to principle of mathematical induction, if P(k+1) = m(k+1) + 5 is true then _____ must be true.
a) P(k) = 3m(k)
b) P(k) = m(k) + 5
c) P(k) = m(k+2) + 5
d) P(k) = m(k)

Answer: b
Clarification: By the principle of mathematical induction, if a statement is true for any number m = k, then for its successor m = k + 1, the statement also satisfies, provided the statement is true for m = 1. So, the required answer is p(k) = mk + 5.

9. Which of the following is the base case for 4n+1 > (n+1)2 where n = 2?
a) 64 > 9
b) 16 > 2
c) 27 < 91
d) 54 > 8

Answer: a
Clarification: Statement By principle of mathematical induction, for n=2 the base case of the inequation 4n+1 > (n+1)2 should be 64 > 9 and it is true.

10. What is the induction hypothesis assumption for the inequality m ! > 2m where m>=4?
a) for m=k, k+1!>2k holds
b) for m=k, k!>2k holds
c) for m=k, k!>3k holds
d) for m=k, k!>2k+1 holds

Answer: b
Clarification: By the induction hypothesis, assume that p (k) = k! > 2k is true, for m=k and we need to prove this by the principle of mathematical induction.

250+ TOP MCQs on Multiplication Theorem on Probability and Answers

Discrete Mathematics Multiple Choice Questions on “Multiplication Theorem on Probability”.

1. How many ways are there to select exactly four clocks from a store with 10 wall-clocks and 16 stand-clocks?
a) 325
b) 468
c) 398
d) 762

Answer: a
Clarification: To choose any clock for the first pick, there are 10+16=26 options. For the second choice, we have 25 clocks left to choose from and so on. Thus, by the rule of product, there are 26 * 25 * 24 * 23 = 650 possible ways to choose exactly four clocks. However, we have counted every clock combination twice. Hence, the correct number of possible ways are 650/2 = 325.

2. If a 12-sided fair die is rolled twice, find the probability that both rolls have a result of 8.
a) (frac{2}{19})
b) (frac{3}{47})
c) (frac{1}{64})
d) (frac{2}{9})

Answer: c
Clarification: Each die roll is independent, that is, if the first die roll result is 8, it will not affect the probability of the second die roll resulting in 8. The probability of rolling one die is (frac{1}{8}). Now, P (1st roll is 8 ∩ 2nd roll is 8). By using the rule of product: (frac{1}{8} * frac{1}{8}). Hence, the probability that both die rolls are 8 is (frac{1}{64}).

3. How many positive divisors does 4000 = 25 53 have?
a) 49
b) 73
c) 65
d) 15

Answer: d
Clarification: Any positive divisor of 4000 must be of the form 2x5y, where x and y are integers satisfying o<=x<=5 and 0<=y<=3. There are 5 possibilities for x and 3 possibilities for y and hence there are 3*5 = 15(rule of product) positive divisors of 4000.

4. Mina has 6 different skirts, 3 different scarfs and 7 different tops to wear. She has exactly one orange scarf, exactly one blue skirt, and exactly one black top. If Mina randomly selects each item of clothing, find the probability that she will wear those clothings for the outfit.
a) (frac{1}{321})
b) (frac{1}{126})
c) (frac{4}{411})
d) (frac{2}{73})

Answer: b
Clarification: There is a (frac{1}{3}) probability that Mina would randomly select the orange scarf, a (frac{1}{6}) probability to select the blue skirt, and a (frac{1}{7}) probability to select the black top. These events are independent, that is, the selection of the scarf does not affect the selection of the tops and so on. Hence, the probability that she selects the clothings of her choice is (frac{1}{3} * frac{1}{6} * frac{1}{7}) = 126.

5. There are 6 possible routes (1, 2, 3, 4, 5, 6) from Chennai to Kochi and 4 routes (7, 8, 9, 10) from the Kochi to the Trivendrum. If each path is chosen at random, what is the probability that a person can travel from the Chennai to the via the 4th and 9th road?
a) (frac{3}{67})
b) (frac{5}{9})
c) (frac{2}{31})
d) (frac{1}{24})

Answer: d
Clarification: There is a (frac{1}{6}) chance of choosing the 4th path, and there is a (frac{1}{4}) chance of choosing the 9th path. The selection of the path to the Kochi is independent of the selection of the path to the Trivendrum. Hence, by the rule of product, there is a (frac{1}{6} * frac{1}{4} = frac{1}{24}) chance of choosing the 4th-9th path.

6. If two 14-sided dice one is red and one is blue are rolled, find the probability that a 3 on the red die, a 5 on the blue die are rolled.
a) (frac{4}{167})
b) (frac{3}{197})
c) (frac{5}{216})
d) (frac{1}{196})

Answer: b
Clarification: Using the rule of product, there are 196 possible combinations of rolls. Since having the red die = 3 and the blue die = 5 are one of the 196 combinations, the required probability is (frac{1}{14}*frac{1}{14} = frac{1}{196}).

7. Suraj wants to go to Delhi. He can choose from bus services or train services to downtown Punjab. From there, he can choose from 4 bus services or 7 train services to head to Delhi. The number of ways to get to Delhi is?
a) 51
b) 340
c) 121
d) 178

Answer: c
Clarification: Since Suraj can either take a bus or a train downtown and he has 4+7=11 ways to head downtown (Rule of sum). After that, he can either take a bus or a train to Delhi and hence he has another 4 * 7 = 11 ways to head to Delhi(Rule of sum). Thus, he has 11 * 11 = 121 ways to head from home to Delhi(Rule of product).

8. Two cards are chosen at random from a standard deck of 52 playing cards. What is the probability of selecting a jack and a Spade from the deck?
a) (frac{4}{13})
b) (frac{1}{13})
c) (frac{4}{13})
d) (frac{1}{52})

Answer: d
Clarification: The required probability is : P(Jack or spade) = P(Jack) * P(spade) = (frac{4}{52} * frac{13}{52} = frac{1}{52}).

9. If I throw 3 standard 7-sided dice, what is the probability that the sum of their top faces equals to 21? Assume both throws are independent to each other.
a) (frac{1}{273})
b) (frac{2}{235})
c) (frac{1}{65})
d) (frac{2}{9})

Answer: a
Clarification: To obtain a sum of 21 from three 7-sided dice is that 3 die will show 7 face up. Therefore, the probability is simply (frac{1}{7} * frac{1}{7} * frac{1}{7} = frac{1}{273}).

10. A box consists of 5 yellow, 12 red and 8 blue balls. If 5 balls are drawn from this box one after the other without replacement, find the probability that the 5 balls are all yellow balls.
a) (frac{5}{144})
b) (frac{6}{321})
c) (frac{4}{67})
d) (frac{1}{231})

Answer: a
Clarification: The total number of the balls in the box is 25. Let events Y: drawing black balls,
R: drawing red balls, B: drawing green balls. Now the balls are drawn without replacement. For the first draw, there are 25 balls to choose from, for the second draw it is 25 − 1 = 24 and 23 for the third draw. Then, the probability that the three balls are all yellow = P(Y1) P(Y2 | Y1) P(Y3 | Y1 ∩ Y2) = (frac{5}{24} * frac{12}{24} * frac{8}{24} = frac{5}{144}).

250+ TOP MCQs on Graphs – Lattices and Answers

Discrete Mathematics Multiple Choice Questions on “Graphs – Lattices”.

1. A Poset in which every pair of elements has both a least upper bound and a greatest lower bound is termed as _______
a) sublattice
b) lattice
c) trail
d) walk

Answer: b
Clarification: A poset in which every pair of elements has both a least upper bound and a greatest lower bound is called a lattice. A lattice can contain sublattices which are subsets of that lattice.

2. In the poset (Z+, |) (where Z+ is the set of all positive integers and | is the divides relation) are the integers 9 and 351 comparable?
a) comparable
b) not comparable
c) comparable but not determined
d) determined but not comparable

Answer: a
Clarification: The two integers 9 and 351 are comparable since 9|351 i.e, 9 divides 351. But 5 and 127 are not comparable since 5 | 127 i.e 5 does not divide 127.

3. If every two elements of a poset are comparable then the poset is called ________
a) sub ordered poset
b) totally ordered poset
c) sub lattice
d) semigroup

Answer: b
Clarification: A poset (P, <=) is known as totally ordered if every two elements of the poset are comparable. “<=” is called a total order and a totally ordered set is also termed as a chain.

4. ______ and _______ are the two binary operations defined for lattices.
a) Join, meet
b) Addition, subtraction
c) Union, intersection
d) Multiplication, modulo division

Answer: a
Clarification: Join and meet are the binary operations reserved for lattices. The join of two elements is their least upper bound. It is denoted by V, not to be confused with disjunction. The meet of two elements is their greatest lower bound. It is denoted by ∧ and not to be confused with a conjunction.

5. A ________ has a greatest element and a least element which satisfy 0<=a<=1 for every a in the lattice(say, L).
a) semilattice
b) join semilattice
c) meet semilattice
d) bounded lattice

Answer: d
Clarification: A lattice that has additionally a supremum element and an infimum element which satisfy 0<=a<=1, for every an in the lattice is called a bounded lattice. A partially ordered set is a bounded lattice if and only if every finite set (including the empty set) of elements has a join and a meet.

6. A sublattice(say, S) of a lattice(say, L) is a convex sublattice of L if _________
a) x>=z, where x in S implies z in S, for every element x, y in L
b) x=y and y<=z, where x, y in S implies z in S, for every element x, y, z in L
c) x<=y<=z, where x, y in S implies z in S, for every element x, y, z in L
d) x=y and y>=z, where x, y in S implies z in S, for every element x, y, z in L

Answer: c
Clarification: A sublattice S of a lattice L is a convex sublattice of L, if x ≤ z ≤ y and x, y in S implies that z belongs to S, for all elements x, y, z in L.

7. Every poset that is a complete semilattice must always be a _______
a) sublattice
b) complete lattice
c) free lattice
d) partial lattice

Answer: b
Clarification: A poset is called a complete lattice if all its subsets have both a join and a meet. Every complete lattice is a bounded lattice. Every poset that is a complete semilattice must always be a complete lattice.

8. A free semilattice has the _______ property.
a) intersection
b) commutative and associative
c) identity
d) universal

Answer: d
Clarification: Any set X may be used to generate the free semilattice FX. The free semilattice is defined to consist of all of the finite subsets of X with the semilattice operation given by ordinary set union; the free semilattice has the universal property.

250+ TOP MCQs on Boolean Algebra – Karnaugh Maps and Answers

Discrete Mathematics Multiple Choice Questions on “Boolean Algebra – Karnaugh Maps”.

1. K-map is used for _______
a) logic minimization
b) expression maximization
c) summing of parity bits
d) logic gate creation

Answer: a
Clarification: K-map(Maurice Karnaugh of Bell labs in 1953) is defined as a diagrammatic method for logic minimization and it is a pictorial view of truth table which shows the relationship between inputs and output. It is more efficient than Boolean algebra. K-map is a diagram made up of squares in which each square represents a minterm or maxterm of the logic function.

2. To display time in railway stations which digital circuit is used?
a) seven segment decoder
b) eight segment encoder
c) 8:3 multiplexer
d) 9 bit segment driver

Answer: a
Clarification: A seven segment decoder is a digital circuit which is used to construct a common type of digital display device i.e., a set of LED (or LCD) segments that display numbers from 0 through 9 at the command of a four-bit code. Moreover, the behavior of the display driver IC is represented by a truth table with seven outputs.

3. Simplify the expression using K-maps: F(A,B,C,D)=Σ (1,3,5,6,7,11,13,14).
a) AB+BC’D+A’B’C
b) BCD’+A’C’D+BD’
c) A’D+BCD+A’BC+AB’C’
d) AC’D’+BC+A’BD+C’D’

Answer: c
Clarification: By solving the given expression we have minterms such as A’D+BCD+A’BC+AB’C’. So, we can get the required expression A’D+BCD+A’BC+AB’C’.

4. When designing a circuit to emulate a truth table, both Product-of-Sums (POS) expressions and Sum-of-Products (SOP) expressions can be derived from?
a) k-map
b) NAND gate
c) NOR gate
d) X-NOR gate

Answer: a
Clarification: A Karnaugh map can be used to build the appropriate POS expression for designing a circuit to form the truth table. Karnaugh maps are not limited to SOP expressions only for minimizing boolean functions.

5. Simplify the expression using K-maps: F(A,B,C) = Σ (1,3,5,6,7).
a) AC’+B’
b) AB+C
c) AB’+B’C’
d) A’BC+B’C+AC

Answer: b
Clarification: By solving the given expression, the minterms are: C and AB. Hence, we can get the required expression C+AB.

6. Simplify the expression using K-maps: F(A,B,C) = π(0,2,4,5,7).
a) (x+y)(y+z)(x+z)(x’+z’)
b) (x+z’)(y+z)(x+y)
c) (x+y’+z)(x+z’)
d) (y’+z’)(x’+y)(z+y’)

Answer: a
Clarification: By solving the given expression, the maxterms are: (x+y), (x’+y), (x+z) and (x’+z’). Hence, we can get required expression (x+y)(x’+y)(x+z)(x’+z’).

7. Addition of two or more bits produces how many bits to construct a logic gate?
a) 108
b) 2
c) 32
d) 64

Answer: b
Clarification: Addition of bits requires carry-in and carry-out bits. Addition of two terms (bits) a and b, and a carry-in bit Cin is required to compute a sum bit S and a carry-out bit Cout. Hence, two bits are produced in general.

8. Use Karnaugh map to find the simplified expression of the function: F = x’yz + xy + xy’z’.
a) xz’+y’z’
b) xy’z+xy
c) y’z+x’y+z
d) yz+xy+xy’z

Answer: d
Clarification: F = x’yz + xyz + xy z’ + xy’z’ is the canonical form for the function. Now, using k-map the minimal form must be: yz+xy+xy’z.

9. Who has invented K-map?
a) Maurice Karnaugh
b) Edward Veitch
c) George Boole
d) Adam Smith

Answer: a
Clarification: The Karnaugh map (KM or K-map) is invented by Maurice Karnaugh in 1953 that is a method of simplifying Boolean expressions.

10. In Gray coding, the adjacent code values differ by _______
a) single bit
b) 3 bits
c) 10 bits
d) 0 bit

Answer: a
Clarification: In Gray coding, the adjacent code values differ only by a single bit. If the given code-word is 01, then the previous and the next code-words are to be 11 or 00 but cannot be 10 in any case. Each cell within a K-map has a definite place-value which is obtained by using this encoding technique. The rows and the columns of the table use Gray code-labeling which in turn represents the values of the corresponding input variables and each K-map cell can be addressed using a unique Gray Code-Word.