Discrete Mathematics and its Applications.pdf - Google Drive

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PG – 1033

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I Semester M.C.A. Examination, January 2015 (Y2K5 Scheme) COMPUTER SCIENCE 1 MCA–4 : Discrete Mathematics and its Applications Time : 3 Hours

Max. Marks : 80

Instruction : Answer the questions in Parts as per the instructions. PART – A Answer any ten questions. Each question carries one mark. 1. a) Construct the truth table for p ∨ ~ p . b) Show that p → (p ∨ q) is a tautology. c) Let f (x) = 3x + 5, g (x) = 2x + 10 ∀ x ∈ Z . Find fog and gof. 3 4 k j d) Find the value of ∑ ( − 3) + ∑ ( −2) . j=0 k=0

e) Define Fibonacci number. f) Prove that nCr + nCr − 1 = n+1Cr . g) If 2nP3 = 2. nP4. Find n. h) If G = (V, E) is the (p, q) graph, show that δ ≤

2q ≤ Δ. p

i) Prove that ∑ d(vi ) = 2 | E | . j) What is the chromatic number of complete bipartite and cycle graph ? k) Show that cube roots of unity from an abelian graph under multiplication. l) Let G = {1, –1, i, – i} be a group under multiplication, find the order of an elements in a graph. P.T.O.

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m) Prove that a ∨ a = a ∀ a ∈ A . n) Define Boolean lattice, give an example. o) Define a cyclic group. PART – B Answer any five full questions, choosing at least two full questions from each Part B and C. Each full question carries 14 marks.

(5×14=70)

2. a) Show that [(p → q) ∧ (q → r) → (p → r ] is a tautology.

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b) Express the definition of limit using quantifiers.

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c) Show that ~ (p → q) ≡ p ∧ ~ q .

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3. a) Prove by mathematical induction that 1+ 3 + 3 + 3 3 + 9 + ... to n terms = 3n 2 − 1 . 3 −1

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b) Show that :

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i) ( Ai ) = (Ai )




ii) ( A i ) =




(A i ) .

4. a) Let n be a positive integer. Then prove that n ∑

k =0

(C)

k n

( −1)

k

n k n 2 Ck = 3n . ∑ and =0 k =0

( )

b) Solve ar + ar – 1 = 3r. 2r. ⎡1 ⎢ ⎢3 A = c) Find A[2] , where ⎢ ⎢⎣ 1

6 4

0 1 1

2⎤ ⎥ 2⎥ ⎥. 0⎥⎦

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5. a) Five men in a company of 20 are graduates. If 3 men are choosen out of 20 at random experiment. Find the probability that

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i) They are all graduates. ii) Atleast 1 is graduate.

b) State and prove Baye’s theorem.

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PART – C 6. a) Prove that in a simple digraph the length of any elementary cycle does not not exceed n.

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b) Define graph isomorphism. Verify whether following graphs are isomorphic to each other.

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7. a) Define Transport network. What are its applications ?

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b) Prove that a connected graph G contains an Eulerian trail iff G has exactly two odd vertices.

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c) Prove that a tree with n nodes has (n – 1) edges. 8. a) State and prove Lagrange’s theorem. b) Define normal subgroup of a group. Prove that the sub group H of G is a normal subgroup of G iff every left coset of H in G is a right coset of H in G. 9. a) For any a and b in a lattice (A, ≤) , prove that a ≤ a ∨ b and a ∧ b ≤ a .

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b) Let L, ∨, ∧ be a finite Boolean algebra, let b∈ L and b ≠ 0, and a1, ...ak be atoms. Such that ai ≤ b , then show that b = a1 ∨ a2 ∨ a2 ∨ ... ∨ ak . ______________

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