## June 14, 2019 Note: We have tried to upload as much as we can, all the question and answers might be shuffled - Please find the answer below each question, some answers might be wrong please review on the last date(some answers might be changed) if you find any wrong answer please comment down below.

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

Question:
Let P(x) be the statement “x + 1 > x.” For the real number domain, qualify the statement for truth value;
Select one:
a. xP(x) is true
b. xP(x) is true
c. xP(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 is always a tautology, 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. xH(x)
d. All of these

Question:
“The sum of two positive integers is always positive” into a logical expression
Select one:
a. BOTH :xy((x > 0) (y > 0) (x +y > 0)) and xy(x +y > 0)
b. xy((x > 0) (y > 0) (x +y > 0))
c. Either xy((x > 0) (y > 0) (x +y > 0)) or xy(x +y > 0)
d. xy(x +y > 0)
The Correct answer is:  BOTH :xy((x > 0)  (y > 0)  (x +y > 0)) and xy(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)  (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)  q

1. 2. 3. 