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Discrete Mathematics Question Paper - Jun 18 - Computer Engineering (Semester 3) - Pune University (PU)
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Discrete Mathematics - Jun 18
Computer Engineering (Semester 3)
Total marks: 80
Total time: 3 Hours
INSTRUCTIONS
(1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Draw neat diagrams wherever necessary.
1.a. Prove :
$1^{3}+2^{3}+3^{3}+\ldots .+n^{3}=\left[\frac{n(n+1)}{2}\right]^{2}$
(4 marks)
00
1.b. Prove that the set of rational numbers is countably infinite.
(4 marks)
00
1.c. Let A = {1, 2, 3} and f1
and f2
are functions from A to
B given by :
$f_{1}=\{(1,2),(2,3),(3,1)\}$ and
$f_{2}=\{(1,2),(2,1),(3,3)\}$
Compute $f_{1}$ o $f_{2}$ and $f_{2} \circ f_{1}$
(4 marks)
00
Or
2.a. Compute the transitive closure of given diagraph using Warshall’s
algorithm :
![enter image description here][1]
(4 marks)
00
2.b. Show that the relation R is “Less than” from A to B
where :
A = {1, 2, 8} and
B = {1, 2, 3, 5}
Find :
(i) R in Roster form
(ii) Domain and Range of R.
(4 marks)
00
2.c. Explain with example, notation used and mathematical expression
to describe the following terms :
(i) Membership
(ii) Subset
(iii) Equality between sets
(iv) Union of sets.
(4 marks)
00
3.a. Write an algorithm for generating permutation of {1, 2, .....
n}. Apply it for n = 3 case.
(4 marks)
00
3.b. Solve the following :
(i) How many different car number plates are possible with
2 letters followed by 3 digits.
(ii) How many of these number plates begin with ‘MH’.
(4 marks)
00
3.c. Consider a graph G(V, E) where V = { $v_{1}, v_{2}, v_{3}$} & deg ($v_{2}$) = 4:
(i) Does such simples graph exists ? If not, why ?
(ii) Does such a multigraph exists ? If yes, give example.
(5 marks)
00
Or
4.a. ) Explain the following in brief : [4]
(i) Subgraphs and spanning subgraph
(ii) Isomorphic graph
(iii) Bipartite graph
(iv) Adjacency matrix and incidence matrix of undirected graph
(5 marks)
00
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