JNTU
JNTU B.Tech II MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE Previous Question Paper (February - 2024 - R13)
Note: i) Question paper consists of Part A, Part B.
ii) Part A is compulsory, which carries 25 marks. In Part A, answer all questions.
iii) In Part B, Answer any one question from each unit. Each question carries 10 marks and may have a, b as sub questions.
PART – A
Give the truth table for the proportional formula (p→q) ↔ (¬p ∨ q).
Answer
| p | q | p → q | ¬p | ¬p ∨ q | (p → q) ↔ (¬p ∨ q) |
|---|---|---|---|---|---|
| T | T | T | F | T | T |
| T | F | F | F | F | T |
| F | T | T | T | T | T |
| F | F | T | T | T | T |
The truth table shows that the formula is a tautology, as the final column consists entirely of true values.
Write the sentence “If 2+3 = 4 then sun rises in the east” in the symbolic form and also write its inverse and converse statement.
Answer
Let p represent the statement "2+3 = 4" and q represent the statement "the sun rises in the east". The symbolic form of the conditional statement is p → q.
The converse statement is q → p, which translates to: "If the sun rises in the east, then 2+3 = 4".
The inverse statement is ¬p → ¬q, which translates to: "If 2+3 ≠ 4, then the sun does not rise in the east".
Define equivalence relation.
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Differentiate between semi group and sub group.
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How many words of three distinct letters can be formed from DUST?
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Give the sequential rule for counting problem.
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What is closed form expression of the sequence an = 4, n ≥ 0?
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Find the coefficient of x32 in (1+x5+x9)10.
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State Euler’s formula for a connected planar graph.
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What are the advantages and disadvantages of breadth first search algorithm?
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PART – B
Show that the following proportions are contradiction:
i) (p ∧ q) ∧ ¬(p ∨ q)
ii) ¬(p ∧ q) ∧ p
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Prove the following logical equivalence without using truth table: ((¬p ∨ ¬q) → (p ∧ q ∧ r)) ⇔ (p ∧ q)
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Describe the rules of inference in predicate logic and verify the validity of the following inference: If one person is more successful than another, then he has worked harder to deserve success. Naveen has not worked harder than Praveen. Therefore Naveen is not more successful than Praveen.
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Explain Warshall’s algorithm with an example.
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Show that the inclusion relation ⊆ is a partial ordering on the power set of a set A.
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If every element of a group G except the identity element is of order 2, then prove that the group G is abelian.
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What are the necessary and sufficient conditions for a homomorphism of a group?
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There are 20 boys and 15 girls in LKG while there are 12 boys and 23 girls in UKG. In how many ways can a team of 10 be chosen so that there are 5 girls and 3 UKG students in the team?
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Enumerate the number of ways of placing 20 indistinguishable balls into 5 boxes where each box is non empty.
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Find the middle term of (x/3 - y/3)9.
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Find the number of integers between 1 and 1000 which are not divisible by 2, 3, 5 or 7.
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Solve an - 6an-1 + 8an-2 = 4n where a0 = 8, a1 = 22.
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Solve the following recurrence relation by substitution: an = an-1 + 3n2, Where a0 = 5.
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Solve an - 2an-1 = 1 for n ≥ 1, where a0 = 2.
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Find a general expression for an using generating functions: an - 3an-1 + 2an-2 = 0, n ≥ 2.
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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 degrees less than 3. Determine the minimum number of vertices G can have.
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Give an example of 2 non isomorphism graphs of order 6 and size 6 with degree sequence (2, 2, 2, 2, 2, 2).
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Find the chromatic number of the following graph (Figure 1).
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Show that the following figure 2 has no Hamiltonian cycle. But the graph has a Hamilton path (Figure 2).
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