[Question Paper] Logic, Discrete Mathematical Structures (Old Course) [January / 2014]
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The document is a question paper for a Logic and Discrete Mathematical Structures course, dated January 2014. It consists of seven questions, with a mix of theoretical and practical problems, covering topics such as mathematical induction, probability, relations, trees, and recurrence relations. Students are required to attempt specific problems and demonstrate their understanding of discrete mathematics concepts.
![L o g i c a n d 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 ( J a n u a r y – 2 0 1 4 ) [ O l 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) Questions No. 1 is Compulsory.
(2) Attempt any four questions from Questions Nos. 2 – 7.
(3) Figure to the right indicate Full Marks.
(4) Each question carries 20 marks.
Q.1 Attempt The Following Question: (20 Marks)
(A) Answer the following in one line:
(i) What is the value of [-9.9]?
(ii) What is inverse of the statement r → 𝑠?
(iii) Write the formula for extended Pigeonhole Principle.
(iv) When p → 𝑞 is true, where p and q are any statements?
(v) If 𝐴 = [
1 0 0
0 1 0
] ; 𝐵 = [
1 0
0 1
0 0
] compute A B.
(5)
(B) Find a minimal spanning tree for the following weighted graph: (5)
(C) What are the values of these sums?
(i) ∑ (𝑘2)5
𝑘=1 (ii) ∑ (−2) 𝑗4
𝑗=0
(5)
(D) Draw the Hasse diagram of the following: –
𝐷30 = {1,2,3,5,6,10,15,30}
(5)](https://image.slidesharecdn.com/ldms-qpoldcoursejan-2014-170717052024/85/Question-Paper-Logic-Discrete-Mathematical-Structures-Old-Course-January-2014-1-320.jpg)
![L o g i c a n d 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 ( J a n u a r y – 2 0 1 4 ) [ O l 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.2 Attempt The Following questions: (15 Marks)
(A) Prove the statement is true by using Mathematical Induction.
12
+ 32
+ 52
+ ⋯ + (2𝑛 − 1)2
=
𝑛(2𝑛 + 1)(2𝑛 − 1)
3
(8)
(B) Solve the recurrence relation 𝑎 𝑛 = 4𝑎 𝑛−1 + 5𝑎 𝑛−2, 𝑎1 = 2, 𝑎2 = 6. (7)
(C) Let 𝑆 = {1,2,3,4,5} and 𝐴 = 𝑆 × 𝑆. Define the following relation R on A: (a, b) R (a’, b’)
if and only if a b’ =a’ b.
(7)
Q.3 Attempt The Following questions: (15 Marks)
(A) Use the structure 𝑅 = [𝑀, +,∗, 𝑇]where M is the set of matrices of the form
[
1 0 0
0 0 0
0 0 0
]. Find for which operation R is closed.
(8)
(B) Two cards are drawn at random from a deck of 52 cards. Find the probability that: –
(i) one is red and other is black
(ii) one is spade and other is heart
(iii) both are face cards
(7)
(C) Let f, g and h are functions form R to R defined as:
𝑓(𝑥) = 2𝑥3
− 7, 𝑔(𝑥) = 3𝑥2
and ℎ(𝑥) = 5𝑥 + 4
Find: –
(i) ((g o f)oh) (1)
(ii) (f o g) (2)
(iii) Verify that (g o f)−1
= 𝑓−1
𝑜𝑔−1
(7)
Q.4 Attempt The Following questions: (15 Marks)
(A) Let R be a relation on A={1,2,3,4} R={(1,1), (1,3), (2,4), (3,1), (3,3), (4,3)}. Find the
transitive closure, by Warshall’s Algorithm for this relation.
(8)
(B) Construct the tree of the following algebraic expression and give the arrays LEFT, DATA
and RIGHT describing the tree. ((2 ∗ 𝑥) + (3 − (4 ∗ 𝑥))) + (𝑥 − (3 ∗ 11))
(7)](https://image.slidesharecdn.com/ldms-qpoldcoursejan-2014-170717052024/85/Question-Paper-Logic-Discrete-Mathematical-Structures-Old-Course-January-2014-2-320.jpg)
![L o g i c a n d 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 ( J a n u a r y – 2 0 1 4 ) [ O l 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 The Following questions: (15 Marks)
(A) Let M be the FSM with following state table: –
a b
𝑆0 𝑆2, 𝑦 𝑆1, 𝑧
𝑆1 𝑆2, 𝑥 𝑆3, 𝑦
𝑆2 𝑆2, 𝑦 𝑆1, 𝑧
𝑆3 𝑆3, 𝑧 𝑆0, 𝑥
(i) Find the input Set I, the state S, the output Set O and the initial state.
(ii) Draw the state diagram D=D(M) of M.
(8)
(B) Let (𝐴, ≤) and (𝐵, ≤)are poset, prove that (𝐴 × 𝐵, ≤) is a poset with partial order ≤
defined by (𝑎, 𝑏) ≤ (𝑎′
, 𝑏′) if 𝑎 ≤ 𝑎′ in B.
(7)
Q.6 Attempt The Following questions: (15 Marks)
(A) If R be a relation on Z defined as (x, y) ∈ 𝑅 iff 3x+5y is divisible by 8. Show that R is an
equivalent relation.
(8)
(B) Solve the following recurrence relations: –
(i) by the method of characteristics roots, 𝑏 𝑛 = 4𝑏 𝑛−1 − 4𝑏 𝑛−2, 𝑏1 = 1, 𝑏2 = 7.
(ii) By method of generating functions 𝑎 𝑛 = 3𝑎 𝑛−1 + 4, 𝑎0 = 5.
(7)
Q.7 Attempt The Following questions: (15 Marks)
(A) Consider the rooted tree (T, a) and answer the question: – (8)](https://image.slidesharecdn.com/ldms-qpoldcoursejan-2014-170717052024/85/Question-Paper-Logic-Discrete-Mathematical-Structures-Old-Course-January-2014-3-320.jpg)
![L o g i c a n d 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 ( J a n u a r y – 2 0 1 4 ) [ O l 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
(i) List all the level four vertices.
(ii) List all the leaves.
(iii) List all siblings of c.
(iv) List offspring of c.
(v) List descendants of c.
(vi) Compute T (f).
(vii) Compute (T, c)?
(viii) What is height of (T, a)?
(ix) What is height of T(f)?
(x) What is the minimal number of vertices that would need to be added to
make (T, a) a complete 3-tree?
(B) Let A={a, b, c, d, e, f, g, h} and let R be the relation defined by: –
(i) Show that (A, R) is a poset.
(ii) Does the poset (A, R) have least element? A greatest element? If so, identify
them.
(iii) Show that the poset (A, R) is complimented and give all pairs of complements.
(iv) Prove or disprove that (A, R) is a Boolean algebra.
(7)](https://image.slidesharecdn.com/ldms-qpoldcoursejan-2014-170717052024/85/Question-Paper-Logic-Discrete-Mathematical-Structures-Old-Course-January-2014-4-320.jpg)
[Question Paper] Logic, Discrete Mathematical Structures (Old Course) [January / 2014]
- 1. L o g i c a n d 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 ( J a n u a r y – 2 0 1 4 ) [ O l 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) Questions No. 1 is Compulsory. (2) Attempt any four questions from Questions Nos. 2 – 7. (3) Figure to the right indicate Full Marks. (4) Each question carries 20 marks. Q.1 Attempt The Following Question: (20 Marks) (A) Answer the following in one line: (i) What is the value of [-9.9]? (ii) What is inverse of the statement r → 𝑠? (iii) Write the formula for extended Pigeonhole Principle. (iv) When p → 𝑞 is true, where p and q are any statements? (v) If 𝐴 = [ 1 0 0 0 1 0 ] ; 𝐵 = [ 1 0 0 1 0 0 ] compute A B. (5) (B) Find a minimal spanning tree for the following weighted graph: (5) (C) What are the values of these sums? (i) ∑ (𝑘2)5 𝑘=1 (ii) ∑ (−2) 𝑗4 𝑗=0 (5) (D) Draw the Hasse diagram of the following: – 𝐷30 = {1,2,3,5,6,10,15,30} (5)
- 2. L o g i c a n d 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 ( J a n u a r y – 2 0 1 4 ) [ O l 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.2 Attempt The Following questions: (15 Marks) (A) Prove the statement is true by using Mathematical Induction. 12 + 32 + 52 + ⋯ + (2𝑛 − 1)2 = 𝑛(2𝑛 + 1)(2𝑛 − 1) 3 (8) (B) Solve the recurrence relation 𝑎 𝑛 = 4𝑎 𝑛−1 + 5𝑎 𝑛−2, 𝑎1 = 2, 𝑎2 = 6. (7) (C) Let 𝑆 = {1,2,3,4,5} and 𝐴 = 𝑆 × 𝑆. Define the following relation R on A: (a, b) R (a’, b’) if and only if a b’ =a’ b. (7) Q.3 Attempt The Following questions: (15 Marks) (A) Use the structure 𝑅 = [𝑀, +,∗, 𝑇]where M is the set of matrices of the form [ 1 0 0 0 0 0 0 0 0 ]. Find for which operation R is closed. (8) (B) Two cards are drawn at random from a deck of 52 cards. Find the probability that: – (i) one is red and other is black (ii) one is spade and other is heart (iii) both are face cards (7) (C) Let f, g and h are functions form R to R defined as: 𝑓(𝑥) = 2𝑥3 − 7, 𝑔(𝑥) = 3𝑥2 and ℎ(𝑥) = 5𝑥 + 4 Find: – (i) ((g o f)oh) (1) (ii) (f o g) (2) (iii) Verify that (g o f)−1 = 𝑓−1 𝑜𝑔−1 (7) Q.4 Attempt The Following questions: (15 Marks) (A) Let R be a relation on A={1,2,3,4} R={(1,1), (1,3), (2,4), (3,1), (3,3), (4,3)}. Find the transitive closure, by Warshall’s Algorithm for this relation. (8) (B) Construct the tree of the following algebraic expression and give the arrays LEFT, DATA and RIGHT describing the tree. ((2 ∗ 𝑥) + (3 − (4 ∗ 𝑥))) + (𝑥 − (3 ∗ 11)) (7)
- 3. L o g i c a n d 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 ( J a n u a r y – 2 0 1 4 ) [ O l 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 The Following questions: (15 Marks) (A) Let M be the FSM with following state table: – a b 𝑆0 𝑆2, 𝑦 𝑆1, 𝑧 𝑆1 𝑆2, 𝑥 𝑆3, 𝑦 𝑆2 𝑆2, 𝑦 𝑆1, 𝑧 𝑆3 𝑆3, 𝑧 𝑆0, 𝑥 (i) Find the input Set I, the state S, the output Set O and the initial state. (ii) Draw the state diagram D=D(M) of M. (8) (B) Let (𝐴, ≤) and (𝐵, ≤)are poset, prove that (𝐴 × 𝐵, ≤) is a poset with partial order ≤ defined by (𝑎, 𝑏) ≤ (𝑎′ , 𝑏′) if 𝑎 ≤ 𝑎′ in B. (7) Q.6 Attempt The Following questions: (15 Marks) (A) If R be a relation on Z defined as (x, y) ∈ 𝑅 iff 3x+5y is divisible by 8. Show that R is an equivalent relation. (8) (B) Solve the following recurrence relations: – (i) by the method of characteristics roots, 𝑏 𝑛 = 4𝑏 𝑛−1 − 4𝑏 𝑛−2, 𝑏1 = 1, 𝑏2 = 7. (ii) By method of generating functions 𝑎 𝑛 = 3𝑎 𝑛−1 + 4, 𝑎0 = 5. (7) Q.7 Attempt The Following questions: (15 Marks) (A) Consider the rooted tree (T, a) and answer the question: – (8)
- 4. L o g i c a n d 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 ( J a n u a r y – 2 0 1 4 ) [ O l 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 (i) List all the level four vertices. (ii) List all the leaves. (iii) List all siblings of c. (iv) List offspring of c. (v) List descendants of c. (vi) Compute T (f). (vii) Compute (T, c)? (viii) What is height of (T, a)? (ix) What is height of T(f)? (x) What is the minimal number of vertices that would need to be added to make (T, a) a complete 3-tree? (B) Let A={a, b, c, d, e, f, g, h} and let R be the relation defined by: – (i) Show that (A, R) is a poset. (ii) Does the poset (A, R) have least element? A greatest element? If so, identify them. (iii) Show that the poset (A, R) is complimented and give all pairs of complements. (iv) Prove or disprove that (A, R) is a Boolean algebra. (7)