This is a Question Papers of Mumbai University for B.Sc.IT Student of Semester - III [Logic and Discrete Mathematics] (Revised Course). [Year - January / 2017] . . . Solution Set of this Paper is Coming soon . . .

1. L o g i c , D i s c r e t e M a t h e m a t i c s
Q u e s t i o n P a p e r ( J a n u a r y – 2 0 1 7 ) [ R e v i s e d C o u r s e ]
1 | Page
M u m b a i B . S c . I T S t u d y
F a c e b o o k | T w i t t e r | I n s t a g r a m | G o o g l e + | Y o u T u b e
– Kamal T.
Time: 3 Hours Total Marks: 100
N.B.: (1) All Questions are Compulsory.
(2) In each question from question No.2 to question No. 7, sub-question (a) is compulsory and attempt any
one from sub-question (b) and (c).
(3) Figure on right indicate maximum marks.
Q.1 Attempt any one of The Question: (10 Marks)
(A) State first principle of finite induction and using it show that:
(i) 12
+ 22
+ 32
+. . . . +𝑛2
=
𝑛(𝑛+1) (2𝑛+1)
6
(ii)
1
3.45
+
2
4.56
+. . . . . +
𝑛
(𝑛+2) (𝑛+3) (𝑛+4)
=
𝑛(𝑛+1)
6(𝑛+3)(𝑛+4)
Note That 3.4 means product of 3 and 4.
(10)
(B) Describe the following with one example and one theorem related to each:
(i) Bipartite Graph (ii) Algebraic Structure
(10)
Q.2 Attempt The Following questions: (15 Marks)
(A) State and prove De Morgan’s Laws for any n sets. (8)
(B) What is Tautology? Compute the truth table of (𝑃 ⟹ 𝑄) ⟺ (~𝑄 ⟹ ~𝑃). Is it
Tautology?
(7)
(C) Out of 240 students in a college, 130 students are in NCC, 110 are in NSS and 80 are in
other activities. 40 are in NCC & NSS both, 35 are in NCC & other activity both and 30
are in NSS & other activity both. 20 students take part in all the three. Prepare Venn
Diagram. Also. Find number of students taking part in atleast one of them, using
inclusion-exclusion principle.
(7)
Q.3 Attempt The Following questions: (15 Marks)
(A) Write a note on Warshall’s Algorithm. Using the algorithm, find 𝑅∞
when 𝐴 =
{1, 2, 3, 4} and 𝑅 = {(1,1), (1,2), (2,3), (3,4)}.
(8)
(B) Draw Hasse diagram of 𝐷20, set of all positive divisors of 20. Is 𝐷20, a poset? Further,
check if 𝐷20 is a lattice.
(7)
(C) Show that any two equivalence classes are equal or disjoint. (7)

2. L o g i c , D i s c r e t e M a t h e m a t i c s
Q u e s t i o n P a p e r ( J a n u a r y – 2 0 1 7 ) [ R e v i s e d C o u r s e ]
2 | Page
M u m b a i B . S c . I T S t u d y
F a c e b o o k | T w i t t e r | I n s t a g r a m | G o o g l e + | Y o u T u b e
– Kamal T.
Q.4 Attempt The Following questions: (15 Marks)
(A) State and prove extended Pigeonhole Principle. Give one example. (8)
(B) What is Binary Operation? Show that binary operation * is associative and
commutative if * is defined as 𝑎 ∗ 𝑏 = 𝑎𝑏/9 for all a and 𝑏 ∈ 𝑄 − {0}. Also, find
identity and inverse element.
(7)
(C) Find fog and gof. Also, check if they are equal when f, g : 𝐼𝑅 → 𝐼𝑅 is defined as 𝑓(𝑥) =
𝑥 + 1 and 𝑔(𝑥) = 𝑥2
for all 𝑥 ∈ 𝐼𝑅.
(7)
Q.5 Attempt The Following questions: (15 Marks)
(A) Write Prim’s Algorithm to find the minimal spanning tree in a graph. (8)
(B) Draw 𝐾6, a complete graph on 6 vertices. Show Hamiltonian cycle in 𝐾6. (7)
(C) Draw Peterson’s Graph. Further, draw three subgraphs of Peterson’s graph. (7)
Q.6 Attempt The Following questions: (15 Marks)
(A) Show that the set 𝑆 = {±1, ± 𝑖, ± 𝑗, ± 𝑘} is an integral domain but not a filed. (8)
(B) Let e : 𝐵2
→ 𝐵6
be an (2.5) encoding function defined as 𝑒(00) − (00000), 𝑒(01) =
11011, 𝑒(11) = 11100 and 𝑒(10) = 00101.
(i) Find minimum distance.
(ii) How many errors can e detect?
(iii) How many errors can e correct?
(7)
(C) Show that 𝐺 = {0, 1, 2, 3, 4, 5} forms an abelian group with respect to addition
modulo 6.
(7)
Q.7 Attempt The Following questions: (15 Marks)
(A) The number of bacteria in a culture is 1000 and this number increases by 250% every
two hours. Using recurrence relation, find number of bacteria present after one day.
(8)
(B) Determine the coefficient 𝑥5
of generating functional (1 − 2𝑥)−7
. (7)
(C) Solve the recurrence relation 𝑎 𝑛 − 3𝑎 𝑛−1 = 5(7 𝑛), where 𝑛 ≥ 1 and 𝑎0 − 2. (7)