JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech II DISCRETE MATHEMATICS Previous Question Paper (April/May -2023 - R18)
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
Define Quantifiers.
Answer
Quantifiers are logical operators used to specify the quantity of elements in a domain that satisfy a given predicate. The Universal Quantifier (∀x P(x)) asserts that P(x) is true for every element x in the domain, while the Existential Quantifier (∃x P(x)) asserts that there exists at least one element x in the domain for which P(x) is true.
Show that
Answer
Using logical equivalences:
By associative and commutative laws:
By De Morgan's law, ¬(p ∧ q) is equivalent to (¬p ∨ ¬q). Since (¬p ∨ ¬q) ∨ r is the definition of (p ∧ q) → r, the expressions are not equivalent unless the original expression was (p ∧ q) → r. Assuming the standard equivalence (p → r) ∧ (q → r) ≡ (p ∨ q) → r, the distributive law confirms the logical structure.
Write properties of lattice.
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Write applications of n-array relations.
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Define structural induction.
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Compare well and partial ordering principle.
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How to calculate coefficient of generating function?
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Find the recurrence relation of the sequence
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Mention applications of a tree.
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Define complete bipartite graph with an example.
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Part B
Discuss important list of implications and equivalences with examples.
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Show that R is logically derived from
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Show that
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Show that the following premises are inconsistent: i. If Jack misses many classes through illness, then he fails high school. ii. If Jack fails high school, then he is uneducated. iii. If Jack reads a lot of books, then he is not uneducated. iv. Jack misses many classes through illness and reads a lot of books.
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Let R be a group of all real numbers under addition and R+ be a group of all positive real numbers under multiplication. Show that the mapping
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Show that the function
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Illustrate the properties of relations with suitable examples.
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If M is set of all non singular matrices of order
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Define Algorithm. Write an algorithm to check whether an element belongs to array or not. Find complexity of algorithm.
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Discuss in detail Pigeonhole principle with an example.
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Find the number of integers between 1 and 250, that are divisible by any of the integers 2, 3, 5, and 7.
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State the strong induction. Prove that a positive integer > 1 is either a prime number or it can be written as product of prime numbers.
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Solve the non-homogenous recurrence relation
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Use the method of generating function to solve the recurrence relation
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How many onto functions are there from a set with six elements to set with three elements?
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Discuss in detail about Linear Non-homogeneous Recurrence Relations with Constant Coefficients with an example.
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Discuss in detail about tree traversal techniques with an example.
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Define Spanning tree and Minimum cost Spanning tree. State and discuss Kruskal's algorithm with an example.
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Give an example of a graph which is i. Eulerian but not Hamilton, ii. Hamilton but not Eulerian, iii. Eulerian and Hamilton, and iv. Neither Hamilton nor Eulerian.
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Draw the complete graph K₅ with vertices A, B, C, D, and E and draw all complete sub-graph of K₅ with 4 vertices.
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