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JNTU B.Tech II MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE Previous Question Paper (March - 2021 - R16)
Answer any five questions. All questions carry equal marks.
Main
Use indirect method to show that P → Q, Q → R, ~(P Λ R), P V R ⇒ R.
Answer
To prove the argument using the indirect method, we assume the negation of the conclusion and show that it leads to a contradiction. The conclusion is R, so we assume ~R is true.
- Assume the premises: (1) P → Q, (2) Q → R, (3) ~(P ∧ R), and (4) P ∨ R.
- Assume the negation of the conclusion: (5) ~R.
- From (2) Q → R and (5) ~R, by Modus Tollens, we get (6) ~Q.
- From (1) P → Q and (6) ~Q, by Modus Tollens, we get (7) ~P.
- From (4) P ∨ R and (7) ~P, by Disjunctive Syllogism, we get (8) R.
- We now have (5) ~R and (8) R, which is a contradiction (R ∧ ~R).
Since the assumption ~R leads to a contradiction, the original argument P → Q, Q → R, ~(P ∧ R), P ∨ R ⇒ R is valid.
Obtain principal conjunctive normal form for the formula p V (~ p → (q V (~q → r)))
Answer
First, simplify the given formula: p ∨ (~p → (q ∨ (~q → r))).
Using the implication rule A → B ≡ ~A ∨ B:
To obtain the Principal Conjunctive Normal Form (PCNF), we express this as a product of maxterms involving all variables p, q, and r. The maxterms are of the form (x ∨ y ∨ z) where each variable is either original or negated.
The expression p ∨ q ∨ r is equivalent to the conjunction of all maxterms except (p ∨ q ∨ r) itself. The maxterms are:
This expression represents the PCNF of the given formula.
Prove that non empty subset H of a group G is a subgroup of G if and only if ab-1 ∈ H, for all a, b ∈ H.
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Let X be the set of all three digit integers i.e X = {x | 100 ≤ x ≤ 999}. If Ai is the set of integers, whose ith digit is i, compute the cardinality of A1 ∪ A2 ∪ A3.
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Solve the recurrence relation: un - 2un-1 + un-2 = 2n.
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Is the following graph planar? If so draw the planar graph.
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Find the minimum spanning tree shown in Graph 2.
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Show that the proposition (p ∨ ¬q) ∧ (¬p ∨ ¬q) ∨ ¬q is a tautology.
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Prove that the argument is valid: No mathematicians are fools. No one who is not a fool is an administrator. Sita is a mathematician. Therefore, Sita is not an administrator.
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The set Q of all rational numbers, the operation is defined by a * b = a + b - ab. Show that under this operation Q forms a commutative monoid.
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Find the coefficient of x4y7 in the expansion of (x - y)11.
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