The document is an examination question paper for a course on logical discrete mathematical structures, dated April 2014. It includes a series of questions covering topics such as DeMorgan's laws, boolean polynomials, equivalence classes, graph isomorphism, minimal spanning trees, and language recognition. Students are required to answer specific questions, with marks allocated to each section.

1. M u m b a i B . S c . I T S t u d y
F a c e b o o k / m u m b a i b s c i t | I n s t a g r a m / k t . b s c i t
Page 1 | L o g i c D i s c r e t e M a t h e m a t i c a l S t r u c t u r e s
Q u e s t i o n P a p e r ( A p r – 2 0 1 4 ) [ O l d C o u r s e ] B Y K a m a l T .
– Kamal T.
Time: 3 Hours Total Marks: 100
N.B.: (1) Questions No. 1 is Compulsory.
(2) Attempt any four questions from question No. 2 to 6.
Q.1 Attempt Both The Question: (10 Marks)
(A) State and prove DeMorgan’s Laws for sets. (5)
(B) Consider the Boolean Polynomial 𝑝(𝑥, 𝑦, 𝑧) = (𝑥 ∧ 𝑦) ∨ (𝑥 ∨ (𝑦′
∧ 𝑧)). Construct
truth table for function 𝑓: 𝐵3 → 𝐵 determined by this polynomial. Draw the logic
diagram.
(5)
Q.2 Attempt Both The Question: (20 Marks)
(A) Write a note on: –
(i) Union of two sets (ii) Complement of a set.
Illustrate using Venn-Diagram.
(10)
(B) What are properties of Mathematical structures? (10)
Q.3 Attempt Both The Question: (20 Marks)
(A) Show that any two equivalence classes are equal or disjoint. (10)
(B) Find 𝑅∞
using Warshall’s Algorithm
𝐴 = {1, 2, 3, 4} and 𝑅 = {(1,1) (1,2) (2,3) (3,4)}
(10)
Q.4 Attempt Both The Question: (20 Marks)
(A) Check if 𝑓(𝑛) and 𝑓(𝑛) have same order.
(i) 𝑓(𝑛) = log(𝑛) and 𝑔(𝑛) = log(𝑛).
(ii) 𝑓(𝑛) = 5𝑛2
+ 4𝑛 + 3 and 𝑔(𝑛) = 𝑛2
+ 100𝑛.
(10)
(B) (i) Determine whether given pair of graph is isomorphic.
(ii) Show that (IR+
, *) is abelian group. IR+
is set of all nonzero real numbers and *
is defined as 𝑎 ∗ 𝑏 =
𝑎𝑏
2
.
(10)

2. M u m b a i B . S c . I T S t u d y
F a c e b o o k / m u m b a i b s c i t | I n s t a g r a m / k t . b s c i t
Page 2 | L o g i c D i s c r e t e M a t h e m a t i c a l S t r u c t u r e s
Q u e s t i o n P a p e r ( A p r – 2 0 1 4 ) [ O l d C o u r s e ] B Y K a m a l T .
– Kamal T.
Q.5 Attempt Both The Question: (20 Marks)
(A) Using Prim’s Algorithm find Minimal Spanning Tree beginning at Vertex E. (10)
(B) Let A=IN and R be relation on A such that xRY if 𝑥 ≤ 𝑦. Show that R is a partial order
and (𝐼𝑁, ≤ is a poset.
(10)
Q.6 Attempt Both The Question: (20 Marks)
(A) Write a note on Languages and concatenation. (10)
(B) Design FSM which recognizes that languages in which every sentence end with 100. (10)