# || - CS201 DISCRETE COMPUTATIONAL STRUCTURES [S3] KTU B TECH MODEL QUESTIONS |QUESTION PAPER PATTERN | IMPORTANT QUESTIONS FOR SECOND YEAR STUDENTS | QUESTION BANK

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**APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY**

__PART D__

__( Answer any TWO, Each carries 9 Marks )__

12.(a) Prove that in a distributive Lattice, if b ⋀ c = 0, then b≤ c. (5 marks)

(b) Show that a ⋁ b is the least upper bound of a and b in (A, ≤).

Show that a ⋀ b is the greatest lower bound of a and b in (A, ≤).(4 marks)

13.Let (H, .) be a subgroup of a Group (G, .) . Let N ={x/xε G, xHx-1 = H}. Show that (N, .) is a subgroup of (G, .).

(9 marks)

14.State and prove Lagrange’s Theorem. (9 marks)

__PART E____( Answer any FOUR, Each carries 10 Marks )__

15.(a)Write the given formula to an equivalent form and which contains the connectives ⏋ and ⋀ only.

⏋ (P⇆ (Q→(R⋁P))) (3 marks)

(b) Show that the following implication is a tautology without constructing the truth table.

((P⋁⏋P) →Q) → ((P⋁⏋P)→R) ⇒ (Q→R) (3 marks)

(c) Show that ( ⏋P⋀ ( ⏋Q⋀R)) ⋁ (Q⋀R) ⋁ (P⋀R) ⇔ R without constructing the truth table. (4 marks)

19.Show that (x) (P(x) ⋁ Q(x)) ⇒ (x)P(x) ⋁ (∃x)Q(x) using Indirect method of Proof. (10 Marks)

17 Discuss Indirect method of Proof. Show that the following premises are inconsistent.

(i) If Jack misses many classes through illness, then he fails high school.

(ii) If Jack fails high School, then he is uneducated.

(iii) If Jack reads a lot of books, then he is not uneducated.

(iv) Jack misses many classes through illness and reads a lot of books. (10 marks)

18.(a) Show that

(i) (∃x) (F(x) ⋀ S(x)) → (y) (M(y) → W(y))

(ii) (∃y) (M(y) ⋀⏋W(y)) the conclusion (x)(F(x) →⏋S(x))

follows. (10 marks)

19.(a) Show that S⋁R is tautologically implied by (P⋁Q) ⋀ (P→R) ⋀ (Q→S) (5 marks)

(b) Show the following implication using rules of inferences

P→ (Q→R),Q→ (R→S) ⇒ P→ (Q→S) (5 marks)

20.Give two translations of each of the following, one using a universal quantifier, and one using an existential quantifier.

(i) All men are human

(ii) No women like John

(iii) [John likes some women

(iv) Some woman likes John

(v) Only women like John.

(vi) Everyone is a man or a woman.

(vii) Some humans are not men.

(viii)If someone is a woman, she likes John. (10 marks)

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