[Question Paper] Logic and Discrete Mathematics (Revised Course) [January / 2017]
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The document is a question paper for a discrete mathematics and logic course, dated January 2017, consisting of several questions across multiple topics. Each question is structured with compulsory sub-questions and offers definitions, proofs, and practical applications of concepts like induction, graph theory, binary operations, and recurrence relations. The total marks for the paper are 100, and it is designed to evaluate the understanding of key mathematical principles and algorithms.
![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)](https://image.slidesharecdn.com/ldm-qprevisedcoursejan-2017-170717110108/85/Question-Paper-Logic-and-Discrete-Mathematics-Revised-Course-January-2017-1-320.jpg)
![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)](https://image.slidesharecdn.com/ldm-qprevisedcoursejan-2017-170717110108/85/Question-Paper-Logic-and-Discrete-Mathematics-Revised-Course-January-2017-2-320.jpg)
[Question Paper] Logic and Discrete Mathematics (Revised Course) [January / 2017]
- 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)