JNTU
JNTU B.Tech II MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE Previous Question Paper (May/June - 2019 - R16)
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.
Part A
Find the negative of p → q.
Answer
The negative of the implication p → q is equivalent to the negation of its logical form, which is ¬(p → q). Since p → q is logically equivalent to ¬p ∨ q, the negation is ¬(¬p ∨ q). By applying De Morgan's Law, this simplifies to p ∧ ¬q.
Test the validity of the following argument: (p ∧ ¬r) → q, ¬q → (p ∧ r) ∴ p ∧ r → r.
Answer
To test the validity of the argument, we analyze the premises and the conclusion using truth tables or logical deduction. The premises are:
- (p ∧ ¬r) → q
- ¬q → (p ∧ r)
The conclusion is (p ∧ r) → r. Note that the conclusion (p ∧ r) → r is a tautology, as it follows the form of the conditional (A ∧ B) → B, which is always true regardless of the truth values of p and r. Since the conclusion is a tautology, the argument is valid regardless of the premises, as there is no case where the premises are true and the conclusion is false.
Alternatively, using formal deduction: the conclusion (p ∧ r) → r is an instance of the law of simplification (conjunction elimination), which states that (p ∧ r) implies r. Since the conclusion is logically true, the argument is valid.
If f(x) = x^2 - y, then find f^-1(y).
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If f: G1 → G2 is a homomorphism and a ∈ G1, then prove that f(a^-1) = [f(a)]^-1.
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How many 5 digit numbers are possible, which are greater than 40000 with the digits 1, 2, 3, 4, 5.
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Find the number of positive integer solutions of x + y + z = 12.
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Solve the recurrence relation u_n - u_{n-1} - 6u_{n-2} = 0.
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Find the generating function of the sequence 1, 3, 3^2, 3^3, ........
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If the adjacency matrix of the Graph is provided, then draw the graph.
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If G is a k regular graph with 18 edges and the order of the graph is 9. Find the value of k.
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Part B
Test the validity of the following argument: If I study, I will not fail in the examination. If I do not watch TV in the evenings, I will study. I failed in the examination. Therefore I must watch TV in the evenings.
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Prove that the following argument is valid: ¬(∃x)(p(x) ∧ q(x)), p(a) ∴ ¬q(a).
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Prove that p ↑ q and ¬(p ∧ q) are logically equivalent.
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Prove that the following argument is valid: ∀x(p(x) → ¬q(x)), ¬(∃x)(r(x) ∨ ¬s(x)) ∧ ¬q(x), p(a) ∴ ¬r(a).
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Let X={1,2,3} and f, g, h and s be functions from X to X given by f={(1,2), (2,3), (3,1)}, g={(1,2), (2,1), (3,3)} h={(1,1), (2,2), (3,1)}. Find fog, fohog.
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If f: G1 → G2 is an isomorphism, then prove that f^-1: G2 → G1 is also an isomorphism.
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Prove that the relation a congruent to b mod H is an equivalence relation.
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Prove that the set of even integers forms a group under addition.
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Find the number of solutions of x1+x2+x3 = 19 with the condition x1 > 1, x2 > 2, x3 > 1.
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Prove that if 11 integers are selected from among {1, 2, … 20}, then the selection includes integer a and b such that a – b = 2.
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Find the number of integers < 250 and divisible by 3 or 5 or 11.
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Suppose 14 students in a class appear at a university examination. Prove that there exists at least two among them whose seat number differ by a multiple of 13.
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Solve the recurrence relation u_n - 2u_{n-1} - 3u_{n-2} = 5n, n ≥ 2, u0 = 1, u1 = 1.
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Solve the recurrence relation using generating function: u_n - 2u_{n-1} + u_{n-2} = 2n, u0 = 2, u1 = 1.
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Suppose that G is a non directed graph with 12 edges. Suppose that G has 6 vertices of degree 3 and the rest have degree less than 3. Determine the minimum number of vertices G can have.
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Find the minimal spanning tree using Kruskal’s algorithm.
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Show that the following graphs are isomorphic.
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Prove that a graph G with at least one edge is 2-chromatic if and only if G has no cycle of odd length.
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