Engineering BE Computer Engineering Sem 3 December 2022 2022 Discrete Structures Graph TheoryDSE Question Paper - Mumbai University | munotes
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Questions asked in this paper
Time: 3 Hours Marks: 80
N.B. : (1) Question Number 1 is compulsory
(2) Solve any three questions from the remaining questions
(3) Make suitable assumptions if needed
(4) Assume appropriate data whenever required. State all assumptions clearly
Q.1 Solve any four of the following questions
a) Prove using Mathematical Induction that 1+5+9+ ....... + (4n-3) = n (2n-1) 5
b) Find the relation set & relation matrix for the following digraph. Determine in degree
& out degree of each vertex. 5
c) State the pigeon hole principle. If 30 people are assembled in a room, then show that
of them must have their birthday on the same day of a week. 5
d) Explain the following terms with suitable example: 5
i) Eulerian graph
e) What is a partial order relation? Determine the hasse diagram for following relation 5
a) What is a transitive closure? Let A={al, a2, a3, a4, a5}. Find the transitive closure of
R using Warshall’s algorithm where relation matrix Me is given as follows- 10
b) What are the isomorphic graphs? Determine whether following graphs are
a) Among the integers 1 to 300, 8
i) How many of them are not divisible by 3 nor by 5 nor by 7?
il) How many of them are divisible only by 3?
b) There are 6 Communication Skills books, 8 Engg. Mathematics books, 10 books on C
Programming. How many ways can be used to choose 2 books of different categories
c) What is a partition set? Determine whether each of the following is a partition. Justify
Let X= {1,2,3,....8,9}. Determine whether or not each of following is a partition
a) What is a group? Let A={5, 10 ,15, 20} 10
i) Prepare the composition table w.r.t. the operation of multiplication modulo 25
ii) Whether it is an abelian group? Justify your answer
iii) Find the inverses of all the elements
iv) Whether it is a cyclic group?
b) What is a ring? Let A= {0, 1, 2, 3, 4, 5}. Determine whether a set A with addition
modulo 6 & multiplication modulo 6 is a commutative ring? Justify your answer. 10
a) Define a lattice. Prove that in a distributive lattice the complement of any element is
unique. Determine whether is a distributive lattice. Find the complements of all
b) Define the term bijective function. Let f: be a function defined by f(x) = 2x-3
Determine whether it is a bijective function. 6
Draw the graph G corresponding to each adjacency matrix
b) What is a planer graph? A connected planer graph has 8 vertices having degrees
2,2,2,3,3,3,4,4. How many edges are there in this graph? 6
c) Write the following statements in a symbolic form using quantifiers. Assume a
i) All students have taken a course in mathematics
il) There is a girl student in a class who is also a sports person
ili) Some students are intelligent, but not hardworking
N.B. : (1) Question Number 1 is compulsory
(2) Solve any three questions from the remaining questions
(3) Make suitable assumptions if needed
(4) Assume appropriate data whenever required. State all assumptions clearly
Q.1 Solve any four of the following questions
a) Prove using Mathematical Induction that 1+5+9+ ....... + (4n-3) = n (2n-1) 5
b) Find the relation set & relation matrix for the following digraph. Determine in degree
& out degree of each vertex. 5
c) State the pigeon hole principle. If 30 people are assembled in a room, then show that
of them must have their birthday on the same day of a week. 5
d) Explain the following terms with suitable example: 5
i) Eulerian graph
e) What is a partial order relation? Determine the hasse diagram for following relation 5
a) What is a transitive closure? Let A={al, a2, a3, a4, a5}. Find the transitive closure of
R using Warshall’s algorithm where relation matrix Me is given as follows- 10
b) What are the isomorphic graphs? Determine whether following graphs are
a) Among the integers 1 to 300, 8
i) How many of them are not divisible by 3 nor by 5 nor by 7?
il) How many of them are divisible only by 3?
b) There are 6 Communication Skills books, 8 Engg. Mathematics books, 10 books on C
Programming. How many ways can be used to choose 2 books of different categories
c) What is a partition set? Determine whether each of the following is a partition. Justify
Let X= {1,2,3,....8,9}. Determine whether or not each of following is a partition
a) What is a group? Let A={5, 10 ,15, 20} 10
i) Prepare the composition table w.r.t. the operation of multiplication modulo 25
ii) Whether it is an abelian group? Justify your answer
iii) Find the inverses of all the elements
iv) Whether it is a cyclic group?
b) What is a ring? Let A= {0, 1, 2, 3, 4, 5}. Determine whether a set A with addition
modulo 6 & multiplication modulo 6 is a commutative ring? Justify your answer. 10
a) Define a lattice. Prove that in a distributive lattice the complement of any element is
unique. Determine whether is a distributive lattice. Find the complements of all
b) Define the term bijective function. Let f: be a function defined by f(x) = 2x-3
Determine whether it is a bijective function. 6
Draw the graph G corresponding to each adjacency matrix
b) What is a planer graph? A connected planer graph has 8 vertices having degrees
2,2,2,3,3,3,4,4. How many edges are there in this graph? 6
c) Write the following statements in a symbolic form using quantifiers. Assume a
i) All students have taken a course in mathematics
il) There is a girl student in a class who is also a sports person
ili) Some students are intelligent, but not hardworking
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