🏫 National Institute of Technology Silchar

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All questions are compulsory.
Question	Details	Marks	CO
1	Let `F(x, y)` be the statement “x can fool y,” where the domain consists of all people in the world. `Use quantifiers` to express each of these statements. (a) Nancy can fool exactly two people. (b) No one can fool both Fred and Jerry.	1 1	CO1
2	Let `Q(x, y)` be the statement “x + y = x - y,” `x ∈ Z` and `y ∈ Z.` Determine the truth value of `∀y∃xQ(x, y).`	1	CO1
3	Anita, Boris, and Carmen are inhabitants of an island “X”. The inhabitants of the island “X” are either knights (who always tell the truth) or knaves (who always lie). What are Anita, Boris, and Carmen if Anita says “I am a knave and Boris is a knight” and Boris says “Exactly one of the three of us is a knight”?	2	CO1
4	(a) Determine the `premise` and `conclusion` in “When I stay up late, it is necessary that I sleep until noon.” and write the `converse` of the statement. (b) Determine the `premise` and `conclusion` in “To be a citizen of this country, it is sufficient that you were born in India” and write the `inverse` of the statement.	3 3	CO1
5	(a) What relevant conclusion or conclusions can be drawn from: “If I take the day off, it either rains or snows.” “I took Tuesday off or I took Thursday off.” “It was sunny on Tuesday.” “It did not snow on Thursday.” (b) Determine whether this argument is valid: “A student in this class has not read the book.” “Everyone in this class passed the first exam.” implies the conclusion “Someone who passes the first exam has not read the book.”	2 2	CO1
6	(a) Prove the correctness of the decryption algorithm in the `RSA algorithm.` (b) Define primitive root modulo prime p. (c) Use `Miller Rabin` test to check 177 is composite or a strong pseudo-prime.	2 2 3	CO4
7	(a) Find x such that `x ≡ -2/11 mod 7.` (b) Compute `φ(121).` (c) Determine the parity bits to be added on the data 11010 in a `Hamming code.`	3 2 3	CO4

Q No.	Question	Marks	CO
1.	Let `F(x, y)` be the statement "x can fool y," where the domain consists of all people in the world. Use `quantifiers` to express each of these statements.? (a) Nancy can fool exactly two people. (b) There is someone who can fool exactly one person besides himself or herself.	1 1	CO1 CO1
2.	Determine the truth value of the statement `∃x∀y(x ≤ y²)` if the domain for the variables consists of (a) The integers. (b) The nonzero real numbers.	1 1	CO1 CO1
3.	(a) Determine whether ∀x(P(x) → Q(x)) and ∀xP(x) → ∀xQ(x) are logically equivalent. Justify your answer (b) Is `[(p ∨ q) ∧ (¬p ∨ r)] → (q ∨ r)` a tautology?	2 2	CO1 CO1
4.	Write each of these statements in the form "if p, then q" in English. (a) A sufficient condition for the warranty to be good is that you bought the computer less than a year ago. (b) Whenever you get a speeding ticket, you are driving over 65 miles per hour (c) It is necessary to have a valid password to log on to the server.	1 1 1	CO1 CO1 CO1
5.	(a) What relevant conclusion or conclusions can be drawn from "If I take the day off, it either rains or snows." "I took Tuesday off or I took Thursday off." "It was sunny on Tuesday." "It did not snow on Thursday." (b) Determine whether this argument is valid: "Somebody in this class enjoys whale watching. Every person who enjoys whale watching cares about ocean pollution. Therefore, there is a person in this class who cares about ocean pollution."	2 2	CO1 CO1
6.	(a) Prove the correctness of the digital signature scheme in `RSA algorithm` (b) Define primitive root. (c) How to check a given number p is prime using `AKS algorithm`	3 2 2	CO4
7.	(a) Find x such that `x ≡ -11/17 mod 21` (b) Compute `φ(49)`? (c) Find the smallest integer x such that x when divided by 4, 7, and 9 gives the remainder 3, 5, and 7 respectively	3 2 3	CO4