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3119
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Reg. No. : ..................................... Name : ..........................................
Third Semester B.Tech. Degree Examination, June 2009 Branch : Computer Science (2003 Scheme) 03-303 : DISCRETE STRUCTURES (RF) Time : 3 Hours
Max. Marks : 100 PART – A
Answer all questions. Each question carries 4 marks. 1. Show that P → (Q → P ) ⇔ ~ P → (P → Q ) without truth table. 2. Define well formed formula, converse and contra positive propositions. 3. What are Relations and Relation Matrices ? 4. Using logical arguments on set membership show that (A – B) – C = A – (B ∪ C), where A, B and C are arbitrary sets. 5. If f : A → B and g : B → C be two functions, show that if g o f is ‘onto’, then ‘g’ is onto. 6. Draw a graph which is Hamiltonian but not Eulerian. Clarify the answer. 7. Show that a set of all polynomials in ‘x’ under the operation of addition is a group. 8. State Lagrange’s Theorem. 9. Show that x3 – 9 is irreducible over F, which is field of integers mod 11. 10. Prove that if in a graph, there is one and only one path between every pair of vertices, then the graph is a Tree. (4×10=40 Marks) P.T.O.
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3119 PART – B
Answer one full question from each Module. Each question carries 20 marks. Module – I 11. a) Prove that (x)(P(x) ∨ Q(x)) ⇒ ( x ) P ( x ) ∨ (∃x ) Q ( x ).
b) If f, g and h be functions from N → N, the set of natural numbers such that f(n) = n + 2, g(n) = 5n and ⎧1 if n is even h (n ) = ⎨ ⎩0 if n is odd
then which of these functions are injective ? Find (fog)oh, fog, goh and hof. 12. a) Show that P → S can be derived from the premises P ∨ Q , Q ∨ R and R → S. b) Prove by Mathematical induction that 72n + (22n–3) (3n–1) is divisible by 25 for all values of natural number ‘n’. Module – II 13. a) Solve : Ur – Ur–1 + 2Ur–2 = r + 2r. b) If H is a subgroup of a group G and N is a normal subgroup of G, show that H ∩ N is a normal subgroup of H. 14. a) How to define chromatic polynomial ? Explain with example. b) Find the generating function of the Fibonacci sequence : F(k + 2) = F(k + 1) + F(k) with F(0) = 1 and F(1) = 1. Module – III 15. a) If U, V are ideal of R, and U + V = {u + v / u ∈ U and v ∈ V}; show that U + V is also an ideal. b) If D is an integral domain a , b ∈ D , with an = b n and am = bm for two relatively prime numbers n and m, then show that a = b. 16. a) Show that every finite group of even order contains at least one element of order ‘2’. b) If U is an ideal of a ring R and 1∈ U, then prove that U = R. ––––––––––––––––
(20×3=60 Marks)