Download JNTUH MCA 1st Sem R17 2018 June-July 841AA Mathematical Foundations Of Computer Science Question Paper

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Code No: 841AA R17

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JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
MCA I Semester Examinations, June/July - 2018
MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE
Time: 3hrs Max.Marks:75

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Note:

This question paper contains two parts A and B.
Part A is compulsory which carries 25 marks. Answer all questions in Part A. Part B consists of 5 Units. Answer any one full question from each unit. Each question carries 10 marks and may have a, b, c as sub questions.

PART - A 5 x 5 Marks = 25

1. Construct the truth table for (P ? Q) ? (P ? Q). [5]

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2. Define complement of a function and inverse of a function. Give examples. [5]
3. Find the coefficient of x⁴y⁹ in the expansion of (x-y)¹³. [5]
4. What is meant by generating function? What is its significance? [5]
5. Find the minimum number of vertices in a simple, connected, planar graph with 19 edges. Justify your answer. [5]

PART -B 5 x 10 Marks = 50

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1. a) Using indirect method of proof, derive P ? S from P ? Q ? R, Q ? ~P, S ? ~R, P.
   b) Contrast propositional logic with predicate logic. [5+5]
   OR
   Explain automatic theorem proving with the following expression P, -P ? (P ? Q) ? Q [10]
2. a) If f: X ? Y and g: Y ? X the function g is equal to f^-1 only if g ° f = I and f ° g = I. Prove the result.
   

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   b) Let f: R ? R and g : R ? R where R is the set of real numbers. Find f ° g and g ° f where f(x) = x² - 2, g(x)=x + 4. State whether these functions are injective, subjective or objective. [5+5]
   OR
   Let L a finite distributive lattice. Then prove that every element in L can be written uniquely (except for order) as the join of irredundant join-irreducible elements. Prove the independent laws for the elements of a lattice. [5+5]
3. a) Find the number of ways three roses, four marigolds and five hibiscuses can be planted:
   i) In a row such that all plants of the same family is next to each other.
   

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   ii) In a row such that the hibiscus are planted in between the other two families of plants. [5+5]
   OR
   a) Find the number of different arrangements of the letters of the word REFERENCE.
   b) State inclusion-exclusion principle. [5+5]
4. Solve the recurrence relation a_n - 7a_n-1 + 26a_n-2 - 24a_n-3 = 0 for n >= 2. [10]
   

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   OR
   Demonstrate the solutions for non-homogeneous recurrence relation. [10]
5. a) Prove that any two simple connected graphs with n vertices and all of degree two are isomorphic.
   b) Suppose G1 and G2 are isomorphic prove that if G1 is connected then G2 is also connected. [5+5]
   OR
   

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   a) State and explain the Four - Colour problem for planar graphs.
   b) Prove that the regions of a planar graph can be 4 - coloured if G has a Hamiltonian cycle. [5+5]

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