This is a Question Papers of Mumbai University for B.Sc.IT Student of Semester - III [Logic and Discrete Mathematics] (Revised Course). [Year - April / 2014] . . .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 ( A p r i l – 2 0 1 4 ) [ 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) Figure to the right indicate the full marks.
Q.1 Attempt Both The Question: (10 Marks)
(A) Answer the following in one line: –
(i) What is the value of f(392) if f is mod 7 function?
(ii) What is Boolean Matrix?
(iii) How to find out the reaction is reflexive from MR?
(iv) What is the range for probability of any event?
(v) When 𝑝 ↔ 𝑞 is true, where p and q are any statements?
(5)
(B) Let 𝐴 = {1, 2, 3, 4, 5, 6} and
𝑃 = {
1 2
4 3
3 4
5 1
5 6
2 6
} be a permutation of A.
(i) Write p as a product of Disjoint Cycles.
(ii) Compute 𝑝2
and 𝑝−1
.
(5)
Q.2 Attempt Any Three From The Following: (15 Marks)
(A) Prove by mathematical induction that the sum of the cubes of three consecutive
integers is divisible by 9.
(5)
(B) In a survey of 260 college students, the following data were obtained:
64 had taken a mathematics course, 94 had taken a computer sciences course, 58 had
taken a Business Course, 28 had taken both a mathematics and business course, 26
had taken both a mathematics and a computer science course, 22 had taken both a
computer science and a business course and 14 had taken all three types of courses.
How many students were surveyed who had taken none of the three of courses?
(5)
(C) Show that the following statements are logically equivalent. (use truth table)
~(𝑝 ↔ 𝑞) ≡ (𝑝⋀~𝑞)⋁(~𝑝⋀𝑞)
(5)
(D) Using that the following statements show that
(i) (𝐴⋃𝐵)′
= 𝐴′⋂𝐵′ (ii) (𝐴⋂𝐵)′
= 𝐴′⋃𝐵′
(5)

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 ( A p r i l – 2 0 1 4 ) [ 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.3 Attempt Any Three From The Following: (15 Marks)
(A) Let A={1, 2, 3, 4}. For the relation R whose matrix is given, find the matrix of the
transitive closure Warshall’s Algorithm.
(5)
(B) If A={a, b, c}. How many relations can be defined on A? How many of them are
equivalence relations?
(5)
(C) Find the relation determined by the diagraph and its matrix. Find indegree &
outdegree of each vertex.
(5)
(D) Let a relation R defined on 𝑍+
as a R b if a/b then prove that (𝑍+
, 𝐼) is a poset. (5)
Q.4 Attempt Any Three From The Following: (15 Marks)
(A) Six friends discover that they have a total of Rs. 2,61 with them on a trip for some
outings show that one or more of them must have at least Rs. 361.
(5)
(B) Find ⌊2.3⌋, ⌈−2.3⌉, ⌊34.67⌋, ⌈−34.67⌉, ⌊2⌋ (5)
(C) Let 𝐴 = 𝐵 = (set of real numbers), f and g are defined as 𝑓(𝑎) = 2𝑎 + 1, 𝑔(𝑏) =
𝑏/3 then verify that (g o f)−1
= 𝑓−1
𝑜 𝑔−1
.
(5)
(D) Explain injective and surjective functions with examples. (5)

3. 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 ( A p r i l – 2 0 1 4 ) [ R e v i s e d C o u r s e ]
3 | 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.5 Attempt Any Three From The Following: (15 Marks)
(A) Decide whether the graph is Eulerian, Hamiltonian or both. (5)
(B) Consider the tree & answer the questions.
(i) Root of the tree, (ii) Siblings of G, (iii) Height of the tree, (iv) List all leaves, (iv)
Descendants of C
(5)
(C) Evaluate the following expression in reverse polish form.
1 2 3 ∧ + 1 2 3 + + −
(5)
(D) Use Prim’s Algorithm to find a minimal spanning tree for the connected graph G given
below. Use E as the initial vertex.
(5)

4. 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 ( A p r i l – 2 0 1 4 ) [ R e v i s e d C o u r s e ]
4 | 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.6 Attempt Any Three From The Following: (15 Marks)
(A) Determine whether the 𝑠𝑒𝑡 together with binary operation is a group. If it is a group,
determine if it is Abelian; specify the identity and the inverse of a generic element.
Where 𝑎 ∗ 𝑏 = 𝑎 + 𝑏 + 2, 𝑎, 𝑏 ∈
(5)
(B) Let Q be the set of rational numbers and define numbers and define 𝑎 ∗ 𝑏 = 𝑎 +
𝑏 – 𝑎𝑏 is (𝑄1 ∗) a monoid? Justify your answer.
(5)
(C) Determine whether the following two graphs G1 & G2 are Isomorphic. (5)
(D) Prove that (F, +, .) is a field, where 𝐹 = {𝑎 + 𝑏√2 | 𝑎, 𝑏 𝑎𝑟𝑒 𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠} (5)
Q.7 Attempt Any Three From The Following: (15 Marks)
(A) Use the techniques of backtracking to find an explicit formula for the sequence defined
by the recurrence relational and initial condition(S).
𝑑 𝑛 = −1.1𝑑 𝑛−1, 𝑑1 = 5
(5)
(B) Find the generating function for 𝑎 𝑘 = 2 + 3𝑘. (5)
(C) Solve the recurrence relation 𝑏 𝑛 = 2𝑏 𝑛−2, 𝑏1 = 1, 𝑏2 = 4. (5)
(D) Solve the recurrence relation by method of generating function 𝑎 𝑛+2 − 5𝑎 𝑛+1 +
6𝑎 𝑛 = 2, 𝑎0 = 1, 𝑎1 = 2.
(5)