Discrete Structures for Computer Science - MCQS

 

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Question:

Let m be an integer with m > 1. R on the set of integers is an equivalence relation if

Select one:

a. {(a, b) | a ≡ b (mod m)}

b. {(a, b) | a ≡ a (mod b)}

c. {(a, b) | b ≡ b (mod a)}

d. {(a, b) | b ≡ a (mod m)}

The Correct answer is: {(a, b) | a ≡ b (mod m)}

Question:

Consider the statement: ∀ x,  y E Z if both xy and x + y are even, then

Select one:

a. both x and y are odd

b. both x and y are even

c. x is even and y is odd

d. x is odd and y is even

The Correct answer is:both x and y are even

Question:

If A = {0, 1}, B = {1, 2}, and C = {0, 1, 2} then what of the following isn’t in A × B × C ?

Select one:

a. (1, 1, 0)

b. (2, 2, 0)

c. (1, 1, 1)

d. (1, 2, 2)

The Correct answer is: (2, 2, 0)

Question:

Let A, B, and C be sets. Identify the the correct one among the following

Select one:

a. None of these

b. A ∩ (B ∩ C) = (C ∪ B) ∪ A

c. A ∪ (B ∩ C) = (C ∪ B) ∩ B

d. A ∩ (B ∪ C) = (C ∩ B) ∩ A

The Correct answer is:  None of these

Question:

Let f1 and f2 be functions from R to R such that f1(x) = x2 and f2(x) = x − x2. What are

the functions f1 + f2 and f1 f2?

Select one:

a. x2 and x4 – x3

b. x and x3 – x2

c. x3 and x2 – x3

d. x and x3 − x4

The Correct answer is: x and x3 − x4

Question:

Determine for what of the following for “f” is not a one-to-one function:

Select one:

a. f  for f (x) = x + 1 from the set of real numbers to itself

b. f  from {a, b, c, d} to {1, 2, 3, 4, 5} with f (a) = 4, f (b) = 5, f (c) = 1, and f (d) = 3

c. f  for f (x) = x2 from Z to the set of integers

d. f  for f (x) = x2 from Z+ to the set of integers

The Correct answer is: f  for f (x) = x2 from Z to the set of integers

Question:

Which of the following is not a logical equivalence for bi-conditional representation ?

Select one:

a. ￢(p → q) ≡ p ∧￢q

b. p ↔ q ≡ (p ∧ q) ∨ (￢p ∧￢q)

c. p ↔ q ≡ ￢p ↔￢q)

d. All mentioned

The Correct answer is: ￢(p → q) ≡ p ∧￢q

Question:

By the second law of de-Morgan ￢(r ∨ s) is equivalent to

Select one:

a. ￢r ∧￢s

b. ￢(￢r ∨￢s)

c. ￢r ∨￢s

d. ￢r ∨ s

The Correct answer is: ￢r ∧￢s

Question:

Let P(x) be the statement “x + 1 > x.” For the real number domain, qualify the statement for truth value;

Select one:

a. ∀x∃P(x) is true

b. ∀xP(x) is true

c. ∃x∀P(x) is true

d. ∃xP(x) is true

The Correct answer is: ∀xP(x) is true

Question:

For {Z+: Z+ < 5} verify if ∀xP(x) holds good for P(x) is “x2 < 10”

Select one:

a. ∀xP(x) is a conjunction

b. None of these

c. ∀xP(x) is false

d. ∀xP(x) holds good

The Correct answer is: ∀xP(x) is false

Question:

Identify correct statement/s among the following

Select one:

a. The relation"Union of sets" is reflexive,but not symmetric

b. The relation"parallel of lines" is always an equivalence relation

c. The relation "Division" is Symmetric

The Correct answer is: The relation"parallel of lines" is always an equivalence relation

Question:

Which of the following is/are true ?

Select one or more:

a. p ∧￢p is always a contradiction

 b. ￢p ∨ p is always a tautology

c. ￢p ∨ p is always a contradiction

d. p ∧￢p is always a tautology

The Correct answer is: ￢p ∨ p is always a tautology, p ∧￢p is always a contradiction

Question:

What of the following expressions does not imply the negation of the proposition, “there is an honest politician” if h(x) represents honesty function:

Select one:

a. ￢∃xH(x)

b. ∃xH(x)

c. ∀x￢H(x)

d. All of these

The Correct answer is: ∃xH(x)

Question:

“The sum of two positive integers is always positive” into a logical expression

Select one:

a. BOTH :∀x∀y((x > 0) ∧ (y > 0) → (x +y > 0)) and ∀x∀y(x +y > 0)

b. ∀x∀y((x > 0) ∧ (y > 0) → (x +y > 0))

c. Either ∀x∀y((x > 0) ∧ (y > 0) → (x +y > 0)) or ∀x∀y(x +y > 0)

d. ∀x∀y(x +y > 0)

The Correct answer is:  BOTH :∀x∀y((x > 0) ∧ (y > 0) → (x +y > 0)) and ∀x∀y(x +y > 0)

Question:

Which of the following is/are statement/s

Select one or more:

a. How hot the day is

b. None of these are statements

c. The temperature is 40 degrees.

d. It is raining in the summer.

The Correct answer is: The temperature is 40 degrees., It is raining in the summer.

Question:

Let P be “you can take the flight,”

Let Q be “you buy a ticket.”

What of the following notates “you can take the flight if and only if you buy a ticket”

Select one:

a. None of these

b. Q →  P

c. P →  Q

d. P ↔ Q

The Correct answer is: P ↔ Q

Question:

Consider: p1 → p2, p2 → p3 and p3 → p1 are true. Then what of the following statements above integer “n” is not equivalent ?

Select one:

a. p3: n2 is even

b. p2: n − 1 is odd

c. None of these

d. p1: n is even

The Correct answer is: None of these

Question:

How can we produce the terms of a sequence if the first 10 terms are 5, 11, 17, 23, 29, 35, 41, 47, 53, 59?

Select one:

a. 6 – 5(n + 1)

b. 5 + 6(n − 1)

c. 5 + 6(n + 1)

d. 6 + 5(n − 1)

The Correct answer is: 5 + 6(n − 1)

Question:

The compound propositions  ￢(p ∨ q) and ￢p ∧￢q are logically equivalent if

Select one:

a. ￢(p ∨ q) ↔ (￢p ∧￢q) is a tautology

b. (p ∨ q) ∨ (￢p ∧￢q) results in identity

c. ￢(p ∨ q) ↔ (￢p ∧￢q) is a contradiction

d. (p ∨ q) ∧ (￢p ∧￢q) is a logical equivalence

The Correct answer is: ￢(p ∨ q) ↔ (￢p ∧￢q) is a tautology

Question:

The proposition (p ∧ q) → (p ∨ q) is not a tautology for

Select one:

a. ≡ (￢p ∨ p) ∨ (￢q ∨ q)

b. ≡ (p ∨q) → (p ∨ q) ≡ ￢(p ∧ q) ∨ (p ∨ q)

c. ≡ (p ∧ q) → (p ∨ q) ≡ ￢(p ∧ q) ∨ (p ∨ q)

d. ≡ (￢p ∨￢q) ∨ (p ∨ q)

The Correct answer is: ≡ (p ∨q) → (p ∨ q) ≡ ￢(p ∧ q) ∨ (p ∨ q)

Question:

Q(x, y) being a statement, for what expression of x and y, the truth values of the propositions Q(1, 2) and Q(3, 0) hold good ?

Select one:

a. y = x + 3

b. x = y + 3

c. y = x - 3

d. x = y - 3

The Correct answer is:  x = y + 3

Question:

Consider the Relation R = {(1,1),(2,3),(3,3),(3,2),(2,2)} on the set A = {1,2,3}.

Which of the following are true w.r.t to R

Select one:

a. R is reflexive but not symmetric and transitive

b. R is reflexive,symmetric and transitive

c. R is reflexive and symmetric only but not transitive

d. R is reflexive and symmetric but not transitive

The Correct answer is:  R is reflexive,symmetric and transitive

Question:

Identify De-Morgan’s law from the following:

Select one:

a. p ∧ (p ∨ q) ≡ p

b. ￢(p ∨ q) ≡ ￢p ∧￢q

c. ￢(p ∧ q) ≡ ￢p ∨ q

d. p ∨ (p ∧ q) ≡ p

The Correct answer is: ￢(p ∨ q) ≡ ￢p ∧￢q