JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD MCA 2018 MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE Previous Question Paper (June/July - 2018 - R17)
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
Construct the truth table for (P ˄ Q) → (P ˅ Q) .
Answer
| P | Q | P ˄ Q | P ˅ Q | (P ˄ Q) → (P ˅ Q) |
|---|---|---|---|---|
| T | T | T | T | T |
| T | F | F | T | T |
| F | T | F | T | T |
| F | F | F | F | T |
Define complement of a function and inverse of a function. Give examples.
Answer
The complement of a function f: A → B, denoted as f', is a function where the output is the negation of the original function's output. For a Boolean function f(x), f'(x) = 1 - f(x). Example: If f(x) = x, then f'(x) = 1 - x.
The inverse of a function f: A → B, denoted as f-1: B → A, exists if f is a bijection (one-to-one and onto). It maps each element y in B back to the unique element x in A such that f(x) = y. Example: If f(x) = x + 2, then f-1(x) = x - 2.
Find the coefficient of x4y7 in the expansion of (x-y)11.
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What is meant by generating function? What is its significance?
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Find the minimum number of vertices in a simple, connected, planar graph with 19 edges. Justify your answer.
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Part B
Using indirect method of proof, derive P → ¬ S from P → Q ˅ R, Q → ¬ P, S → ¬ R, P.
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Contrast propositional logic with predicate logic.
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Explain automatic theorem proving with the following expression P, ¬ P ˅ ( P ˅ Q) → Q
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If f : X→Y and g : Y→X the function g is equal to f-1 only if g • f = Ix and f •g = Iy .Prove the result.
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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.
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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.
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Prove the independent laws for the elements of a lattice.
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Find the number of ways three roses, four marigolds and five hibiscuses can be planted:
In a row such that all plants of the same family is next to each other.
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In a row such that the hibiscus are planted in between the other two families of plants.
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Find the number of different arrangements of the letters of the word REFERENCE.
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State inclusion-exclusion principle.
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Solve the recurrence relation an - 7an-1 + 26an-2 - 24an-3 = 0 for n >= 2.
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Demonstrate the solutions for non-homogeneous recurrence relation.
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Prove that any two simple connected graphs with n vertices and all of degree two are isomorphic.
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Suppose G1 and G2 are isomorphic prove that if G1 is connected then G2 is also connected.
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State and explain the Four - Colour problem for planar graphs.
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Prove that the regions of a planar graph can be 4 - coloured if G has a Hamiltonian cycle.
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