MA6566 - Discrete Mathematics -...

MA6566 - Discrete Mathematics

UNIT I

1. Check the validity of the following argument. “If the band could not play rock music or the refreshments were not delivered on time, then the New year’s party would have been cancelled and Alice would have been angry. If the party were cancelled, then refunds would have to be made. No refunds were made”.

(ii) Show that R S� follows logically from the premises C D,(C D) H, H (A B)� � o� � o �� and � �(A B) R S .�� o � 2. Translate the following predicate calculus formula into English sentence

� � � � � � x[C x

3. Prove that � � � � � �p q p r q r r� � o � o oª º¬ ¼r) q r .� � � �

UNIT II

1. Show that ( 1)( 2)( 3)1.2.3 2.3.4 3.4.5 . . . ( 1)( 2) , 1.

4n n n nn n n n� � �

� � � � � � t

2 3n n� �� is divisible by 7, 0nt . 5. Prove the principle of inclusion-exclusion using mathematical induction 6. Find the number of positive integers not exceeding 100 that are divisible by 5 or 7.

UNIT III

1. Define Eulerian graph and Hamiltonian graph. Give an example of (I) a graph which is Hamiltonian but not Eulerian. (II) a graph which is Eulerian but not Hamiltonian. (III) a graph which is both Eulerian and Hamiltonian. (IV) a graph which is non Eulerian and non Hamiltonian

� �

y C(y F x,y ].� �� Here C(x) : x has a computer, F(x ,y) : x and y are friends. The universe for both x and y is the set of all students of your college.

is a tautology.

4. Find the PDNF and PCNF of �p q� ( p � �

5. Check whether the hypothesis “ It is not sunny this afternoon and it is colder than yesterday”, “ we will go swimming only if it is sunny” , “If we do not go swimming then we will take a canoe trip” and “ If we take a canoe trip, then we will be home by sunset” lead to the conclusion “we will be home by sunset”.

2. Using generating function method to solve the Fibonacci series 3. Solve s(k) – 10 s(k-1) + 9 s(k-2) = 0 with s(0) = 3 , s(1) = 11. 4. Using mathematical induction show that 2 2 1

www.studentsfocus.com

2. Check the given graph is strongly connected, weakly connected and unilaterally connected or not. If G is a simple graph with n- vertices and k- components then

the no.of

edges is atmost 2

)1()( �� knkn

3. A connected graph G is Eulerian if and only if every vertex of G is of even degree. Find the Hamiltonian path and Hamiltonian cycle, if it exist. Also identify the Euler circuit

5. Let G be a (p, q) graph. Let M be the maximum degree of the vertices of G and let m be the

minimum degree of the vertices of G. Show that Mpqm dd

2 .

UNIT IV

1. State and prove Fundamental theorem on homomorphism of groups Let ( G , * ) be a finite cyclic group generated by an element a � G. If G is of order n,prove that an = e and G = { a , a2 , …, an = e} where n is the least positive integer for which an = e.

2. Prove that every finite group of order n is isomorphic to a permutation group of degree n.

4. Check the given two graphs G and G’ are Isomorphic or not.


3. Let (G ,H , * ) be a group and a � G,. Let f : G o G be given by f(x) = a*x*a-1 for every x� G prove that f is an isomorphism of G onto G.

4. State and prove Lagranges Theorem 5. The intersection of any two normal subgroup of a group is a normal subgroup. 6. A subgroup H of G is normal iff each left coset of H in G is equal to the right coset of H in

G.

UNIT V

1. In a Lattice show that a d b and c d d implies a *c d b * d. 2. Let ( L, d ) be a lattice. For any a, b � L, a d b � a � b = a � a � b = b. 3. In a distributive lattice prove that a *b = a * c and a � b = a � c implies that b = c 4. Show that every distributive lattice is modular. Whether the converse is true? Justify

Your claim 5. Establish De.Morgan’s laws in a Boolean Algebra. 6. State and prove distributive inequalities of a Lattice 7. Show that in a complemented and distributed lattice

* ' 0 ' 1 ' 'a b a b a b b ad � � � � d


1. Show that ( )p q r� � and ( ) ( )p q p r� � � are logically equivalent. (N/D 2012)

2. Without using the truth table, prove that � � � �p q r q p r� o o { o � .

(N/D 2010)

5. Prove that � � � � � � p q q r p ro � o � o . (M/J 2014)

x PCNF and PDNF

6. Without using truth table find the PCNF and PDNF of

� � � �� P Q P P Q Ro � � � o � �� . (A/M 2011)

7. Find the principal disjunctive normal form of the statement,

� �� q p r p r q� � � � � . (N/D 2012)

3. Prove

x Simplification by Truth Table and without Truth Table

�� p� q� � ��p� ��q� �r �� p��q� � ��p��r � is a tautology.

(N/D 2013)

4. Prove that � � � � � �PoQ � RoQ P R Q� � o . (M/J 2013)

SUBJECT NAME : Discrete Mathematics

SUBJECT CODE : MA2265

MATERIAL NAME : University Questions

REGULATION : R2008

Unit – I (Logic and Proofs)


8. Obtain the principal disjunctive normal form and principal conjunction form of the

statement � �� p p q q r� � o � � o . (N/D 2010)

9. Show that � � � � � � � �P R Q P P Q R P Q R� o � l � � � ��

� � � � � �P Q R P Q R P Q R�� . (M/J 2013)

x Theory of Inference

10. Show that: � � � � � � � � � �, , P Q R S Q M S N M No � o � � o � � and

11. Prove that the following argument is valid: , , p q r q r po� o � � .

(M/J 2012)

12. Prove that the premises ( ), ( )a b c d b co o o �� and ( )a d� are inconsistent.

(N/D 2010)

13. Using indirect method of proof, derive p so� from the premises

14. Prove that A Do� is a conclusion from the premises A B Co � , B Ao� and D Co� by using conditional proof. (M/J 2014)

16. Determine the validity of the following argument:

If 7 is less than 4, then 7 is not a prime number, 7 is not less than 4. Therefore 7 is a prime number. (M/J 2012)

17. State and explain the proof methods. (M/J 2014)

18. Prove that 2 is irrational by giving a proof using contradiction. (N/D 2011),(M/J 2013),(N/D 2013)

x Quantifiers

po�q� r � , ,qo�p so�r and p . (N/D 2011)

15. Show that the hypothesis, “It is not sunny this afternoon and it is colder than yesterday”, “we will go swimming only if it is sunny”, “If we do not go swimming, then we will take a canoe trip” and “If we take a canoe trip, then we will be home by sunset” lead to the conclusion “We will be home by sunset”. (N/D 2012),(N/D 2013)


20. Use the indirect method to prove that the conclusion ( )zQ z� follows form the

premises � �( ) ( )x P x Q x� o and ( )yP y� . (N/D 2012)

21. Prove that � � � � � � � �( ) ( ) ( ) ( )x P x Q x x P x x Q x� � � � � � . (M/J 2013)

22. Prove that � � � � � �( ) ( ) , ( ) ( ) ( ) ( )x P x Q x x R x Q x x R x P x� o � o� � � o� .

(N/D 2010)

23. Use indirect method of proof to prove that

� � � � � � � �( ) ( ) ( ) ( )x P x Q x x P x x Q x� � � � � � . (A/M 2011),(N/D 2011)

25. Show that � � � � � �( ) ( ) ( ) ( )x P x xQ x x P x Q x� o� � o . (M/J 2014)

26. Show that the statement “Every positive integer is the sum of the squares of three integers” is false. (N/D 2011)

28. Verify that validating of the following inference.

Unit – II (Combinatorics)

x Mathematical Induction and Strong Induction

1. Prove by the principle of mathematical induction, for ' 'n a positive integer,

2 2 2 2 ( 1)(2 1)1 2 3 ...6

n n nn � �� . (M/J 2012)

2. Use Mathematical induction show that � � � �2

1

1 2 16

n

k

n n nk

� � ¦ . (A/M 2011)

24. Show that ��x� �P(x)�Q(x)�� x�P(x)� ��x�Q(x) . Is the converse true?

(N/D 2013)

If one person is more successful than another, then he has worked harder to deserve success. Ram has not worked harder than Siva. Therefore, Ram is not more successful than Siva. (A/M 2011)

27. Verify the validity of the following argument. Every living thing is a plant or an animal. John’s gold fish is alive and it is not a plant. All animals have hearts. Therefore John’s gold fish has a heart. (M/J 2012)


3. Using mathematical induction to show that 1 1 1 1... , 21 2 3

n nn

� � � � ! t .

(N/D 2011)

4. Prove by mathematical induction that 2 2 16 7n n� �� is divisible by 43 for each positive integer n . (N/D 2013)

5. Prove, by mathematical induction, that for all 31, 2n n nt � is a multiple of 3.

(N/D 2010)

7. Prove that the number of subsets of set having n elements is 2n . (M/J 2014)

9. State the Strong Induction (the second principle of mathematical induction). Prove that a positive integer greater than 1 is either a prime number or it can be written as product of prime numbers. (M/J 2013)

x Pigeonhole Principle

10. If n Pigeonholes are occupied by � �1kn� pigeons, where k is positive integer, prove

two of the m integers are even. (N/D 2011)

11. What is the maximum number of students required in a discrete mathematics class to be sure that at least six will receive the same grade if there are five possible grades A, B, C, D and F? (N/D 2012)

x Permutations and Combinations

12. How many positive integers n can be formed using the digits 3,4,4,5,5,6,7 if n has to exceed 5000000? (N/D 2010)

13. Find the number of distinct permutations that can be formed from all the letters

of each word (1) RADAR (2) UNUSUAL. (M/J 2012)

that at least one Pigeonhole is occupied by k � 1 or more Pigeons. Hence, find the

minimum number of m integers to be selected from S ^1,2, ...,9` so that the sum of

8. Let m any odd positive integer. Then prove that there exists a positive integer n such

that m divides 2 1n � . (M/J 2013)

6. Use Mathematical induction to prove the inequality n � 2n for all positive integer n . (N/D 2012)


14. A box contains six white balls and five red balls. Find the number of ways four balls can

be drawn from the box if

(1) They can be any colour

(2) Two must be white and two red

(3) They must all be the same colour (A/M 2011)

x Solving recurrence relations by generating function

15. Find the generating function of Fibonacci sequence. (N/D 2013)

16. Using the generating function, solve the difference equation

2 1 1 06 0, 1, 2n n ny y y y y� �� . (N/D 2010)

17. Using generating function solve 2 15 6 0, 0n n ny y y n� �� t with 0 1y and 1 1y .

(A/M 2011)

18. Using method of generating function to solve the recurrence relation

1 24 4 4 ; 2nn n na a a n� � � � t , given that 0 2a and 1 8a . (N/D 2011)

19. Use generating functions to solve the recurrence relation

1 23 4 0, 2n n na a a n� �� t with the initial condition 0 13, 2a a � .

(N/D 2012)

20. Using generating function, solve the recurrence relation 1 25 6 0n n na a a� �� where

02, 0n at and 1 1a . (M/J 2013)

21. Solve the recurrence relation 13 2, 1n na a n� � t , with 0 1a by the method of

generating functions. (M/J 2014)

22. Solve the recurrence relation 21 03 , 0, 3n na a n n n a� � � t . (N/D 2011)

23. Solve the recurrence relation, ( ) ( 1) 2( 1)S n S n n � � � , with (0) 3, (1) 1S S ,

by finding its generating function. (M/J 2012)

24. Solve the recurrence relation 1 2 33 3n n n na a a a� � � � � � given that 0 15, 9a a

and 2 15a . (M/J 2014)

25. A factory makes custom sports cars at an increasing rate. In the first month only one car is made, in the second month two cars are made and so on, with n cars made in the thn month. (N/D 2013)


(i) Set up recurrence relation for the number of cars produced in the first n

months by this factory.

(ii) How many cars are produced in the first year?

x Inclusion and Exclusion

26. Find the number of integers between 1 and 250 both inclusive that are divisible by any of the integers 2,3,5,7. (N/D 2010)

27. Determine the number of positive integers , 1 2000n nd d that are not divisible by

2,3 or 5 but are divisible by 7. (M/J 2013)

28. Find the number of positive integers 1000d and not divisible by any of 3, 5, 7

and 22 . (M/J 2014)

(1) How many of these 2500 students have taken a course in any of these three courses C, Pascal and Networking?

(2) How many of these 2500 students have not taken a course in any of these three courses C, Pascal and Networking? (A/M 2011)

31. Suppose that there are 9 faculty members in the mathematics department and 11 in the computer science department. How many ways are there to select a committee to develop a discrete mathematics course at a school if the committee is to consist of three faculty members form the mathematics department and four from the computer science department? (N/D 2012)

30. A total of 1232 students have taken a course in Spanish, 879 have taken a course in French, and 114 have taken a course in Russian. Further, 103 have taken courses in both Spanish and French, 23 have taken courses in both Spanish and Russian, and 14 have taken courses in both French and Russian. If 2092 students have taken atleast one of Spanish, French and Russian, how many students have taken a course in all three languages? (N/D 2013)

29. There are 2500 students in a college, of these 1700 have taken a course in C, 1000 have taken a course Pascal and 550 have taken a course in Networking. Further 750 have taken courses in both C and Pascal. 400 have taken courses in both C and Networking, and 275 have taken courses in both Pascal and Networking. If 200 of these students have taken courses in C, Pascal and Networking.


Unit – III (Graph Theory)

x Drawing graphs from given conditions

1. Draw the complete graph 5K with vertices , , , ,A B C D E . Draw all complete sub graph

of 5K with 4 vertices. (N/D 2010)

2. Draw the graph with 5 vertices, , , , ,A B C D E such that deg( ) 3A , B is an odd

vertex, deg( ) 2C and D and E are adjacent. (N/D 2010)

3. Draw the graph with 5 vertices , , ,A B C D and E such that deg( ) 3A , B is an odd

vertex, deg( ) 2C and D and E are adjacent. (A/M 2011)

4. Determine which of the following graphs are bipartite and which are not. If a graph is bipartite, state if it is completely bipartite. (N/D 2011)

5. Find the all the connected sub graph obtained from the graph given in the following Figure, by deleting each vertex. List out the simple paths from A to in each sub graph. (A/M 2011)

x Isomorphism of graphs

6. Determine whether the graphsG andH given below are isomorphic. (N/D 2012)


7. Using circuits, examine whether the following pairs of graphs 1 2,G G given below are

isomorphic or not: (N/D 2011)

8. Examine whether the following pair of graphs are isomorphic. If not isomorphic, give the reasons. (A/M 2011)

9. Check whether the two graphs given are isomorphic or not. (M/J 2013)

10. Determine whether the following graphs G and H are isomorphic. (N/D 2013)


Engineering Mathematics 2014

11. The adjacency matrices of two pairs of graph as given below. Examine the isomorphism

of G and H by finding a permutation matrix.

0 0 1 0 1 10 0 1 , 1 0 01 1 0 1 0 0

G H« » « »

(N/D 2010)

x General problems in graphs

12. How many paths of length four are there froma andd in the simple graph G given below. (N/D 2012)

13. Find an Euler path or an Euler circuit, if it exists in each of the three graphs below. If it does not exist, explain why? (N/D 2011)

14. Which of the following simple graphs have a Hamilton circuit or, if not, a Hamilton path?

A Aª º ª º

« » « »« » « »


15. Check whether the graph given below is Hamiltonian or Eulerian or 2-colorable. Justify your answer. (M/J 2013)

x Theorems

16. Prove that an undirected graph has an even number of vertices of odd degree. (N/D 2012),(M/J 2014)

17. Show that if a graph with n vertices is self-complementary then 0 or 1 (mod 4)n { . (M/J 2013)

18. Prove that the maximum number of edges in a simple disconnected graphG with n

vertices and k components is ( )( 1)

2n k n k� � �

. (N/D 2011)

19. Prove that a simple graph with n vertices and k components cannot have more than ( )( 1)

2n k n k� � �

edges. (N/D 2013)

20. Show that graph G is disconnected if and only if its vertex set V can be partitioned into two nonempty subsets 1V and 2V such that there exists no edge is G whose one end

vertex is in 1V and the other in 2V . (M/J 2012)


21. If all the vertices of an undirected graph are each of degree k , show that the number of

edges of the graph is a multiple of k . (N/D 2010)

22. Let G be a simple undirected graph with adjacency matrix A with respect to the ordering 1 2 3, , , ..., nv v v v . Prove that the number of different walks of length r from iv

to jv equals the ( , )i j th entry of rA , where r is a positive integer. (M/J 2013)

x Theorems based on Euler and Hamilton graph

23. Prove that a connected graph G is Eulerian if and only if all the vertices are of even degree. (M/J 2012),(N/D 2013),(M/J 2014)

24. Show that the complete graph with n vertices nK has a Hamiltonian circuit whenever

3n t . (N/D 2012)

25. Prove that if G is a simple graph with at least three vertices and ( )

(G)2

V GG t

thenG is Hamiltonian. (M/J 2013)

26. Let G be a simple undirected graph withn vertices. Let u and v be two non adjacent vertices in G such that deg( ) deg( )u v n� t in G . Show that G is Hamiltonian if and

only if G uv� is Hamiltonian. (A/M 2011)

Unit – IV (Algebraic Structures)

x Group, Subgroup and Normal Subgroup

1. If � �,*G is an abelian group, show that � �2 2 2* *a b a b . (N/D 2010)

2. If * is a binary operation on the set R of real numbers defined by * 2a b a b ab � � ,

(1) Find ,*R is a semigroup

(2) Find the identity element if it exists

(3) Which elements has inverse and what are they? (A/M 2011)

3. Find the left cosets of the subgroup > @ > @^ `0 , 3H of the group 6 6,z �ª º¬ ¼ .

(M/J 2014)


4. State and prove Lagrange’s theorem.

(N/D 2010),(A/M 2011),(M/J 2012),(N/D 2013),(M/J 2014)

5. Prove that the order of a subgroup of a finite group divides the order of the group. (N/D 2011),(M/J 2013)

6. Find all the subgroups of � �9 9,z � . (M/J 2014)

7. Prove the theorem: Let ,*G be a finite cyclic group generated by an element a G� .

If G is of order n , that is, G n , then na e , so that ^ `2 3, , , ..., nG a a a a e

a e . (N/D 2011)

8. Prove that intersection of any two subgroups of a group � �,*G is again a subgroup of

� �,*G . (N/D 2013)

9. Prove that intersection of two normal subgroups of a group � �,*G is a normal

subgroup of a group � �,*G . (M/J 2013)

10. Prove that every cyclic group is an abelian group. (N/D 2013)

11. Prove that the necessary and sufficient condition for a non empty subset H of a group

^ `,*G to be a sub group is 1, *a b H a b H�� . (N/D 2012)

12. If * is the operation defined on S Q Q u , the set of ordered pairs of rational

numbers and given by � � � � � �, * , ,a b x y ax ay b � , show that � �,*S is a semi group.

Is it commutative? Also find the identity element of S . (N/D 2012)

13. Define the Dihedral group 4 ,*D and give its composition table. Hence find the identify

element and inverse of each element. (A/M 2011)

x Homomorphism and Isomorphism

14. Prove that every finite group of order n is isomorphic to a permutation group of order n . (N/D 2011),(M/J 2013)

15. State and prove the fundamental theorem of group homomorphism. (N/D 2013)

16. Let :f G Gco be a homomorphism of groups with Kernel K . Then prove that K is a

normal subgroup of G and /G K is isomorphic to the image of f . (M/J 2012)

.

Further more n is a least positive integer for which n


17. Let � �,*G and � �,H ' be two groups and � � � �: ,* ,g G Ho ' be group

homomorphism. Then prove that the Kernel of g is normal subgroup of � �,*G .

(M/J 2013)

18. Show that the Kernel of a homomorphism of a group ,*G into an another group

,H ' is a subgroup of G . (A/M 2011)

19. If :f G Gco is a group homomorphism from ^ `,*G to ^ `,Gc ' then prove that for

any � � > @ 11, ( )a G f a f a �� . (N/D 2012)

20. If � �,Z � and � �,E � where Z is the set all integers and E is the set all even integers,

show that the two semi groups � �,Z � and � �,E � are isomorphic. (N/D 2010)

21. Let � �,*S be a semigroup. Then prove that there exists a homomorphism : Sg S So ,

where ,SS is a semigroup of functions from S to S under the operation of (left)

composition. (N/D 2011)

22. Let � �,*M be a monoid. Prove that there exists a subset MT M� such that � �,*M

is isomorphic to the monoid � �,T ; here MM denotes the set of all mappings from

M to M and '' '' denotes the composition of mappings. (M/J 2014)

x Ring and Fields

23. Show that � �, ,Z � u is an integral domain where Z is the set of all integers.

(N/D 2010)

24. Prove that the set > @ > @ > @ > @^ `4 0 , 1 , 2 , 3Z is a commutative ring with respect to the

binary operation addition modulo and multiplication modulo 4 4,� u . (N/D 2012)

Unit – V (Lattices and Boolean algebra)

x Partially Ordered Set (Poset)

1. Show that � �,N d is a partially ordered set where N is set of all positive integers and

d is defined by m nd iff n m� is a non-negative integer. (N/D 2010)


2. Draw the Hasse diagram for (1) ^ `1 2,3,6,12,24P (2) ^ `2 1,2,3,4,6,12P and d

is a relation such x yd if and only is |x y . (A/M 2011)

3. Draw the Hasse diagram representing the partial ordering � �^ `, :A B A B� on the

power set ( )P S where ^ `, ,S a b c . Find the maximal, minimal, greatest and least

elements of the poset. (N/D 2012)

x Lattices

4. Show that in a lattice is a bd and c dd , then * *a c b dd and a c b d� d � . (M/J 2014)

5. In a distributive lattice prove that * *a b a c and a b a c� � imply b c . (M/J 2014)

6. Let L be lattice, where * glb(a,b)a b and lub(a,b)a b� for all ,a b L� . Then

both binary operations * and � defined as in L satisfies commutative law, associative law, absorption law and idempotent law. (M/J 2013)

7. In a distributive Lattice ^ `, ,L � � if an element a L� a complement then it is unique.

(N/D 2012)

8. Show that every chain is a lattice. (M/J 2013)

9. Prove that every chain is a distributive lattice. (N/D 2013)

10. Prove that every distributive lattice is modular. Is the converse true? Justify your claim. (A/M 2011)

11. Show that in a distributive and complemented lattice * 0 1 a b a b a b b ac c c cd � � � � d . (M/J 2013)

12. Show that in a lattice if a b cd d , then (N/D 2013)

(i) *a b b c�

(ii) � � � � � � � �* * *a b b c b a b a c� � �

13. Show that the direct product of any two distributive lattices is a distributive lattice. (A/M 2011) ,(M/J 2012)


14. If ( )P S is the power set of a set S and ,� � are taken as join and meet, prove that

( ),P S � is a lattice. Also, prove the modular inequality of a Lattice ,L d for any

, ,a b c L� , � � � �a c a b c a b cd � � � d � � . (N/D 2011)

15. Prove that Demorgan’s laws hold good for a complemented distributive lattice

, ,L � � , viz � �a b a bc c c� � and � �a b a bc c c� � . (N/D 2011),(M/J 2013)

16. Establish de Morgan’s laws in a complemented, distributive lattice. (M/J 2014)

17. If 42S is the set all divisors of 42 and D is the relation “divisor of” on 42S , prove that

^ `42 ,S D is a complemented Lattice. (N/D 2010)

x Boolean Algebra

18. In a Boolean algebra, prove that � �a b a bc c c� � . (N/D 2010)

19. In any Boolean algebra, show that 0ab a bc c� if and only if a b . (N/D 2011)

20. In any Boolean algebra, prove that the following statements are equivalent:

(1) a b b� (2) a b a (3) 1a bc � and (4) 0a bc (N/D 2011)

21. In a Boolean algebra, prove that .( )a a b a� , for all ,a b B� . (N/D 2012)

22. Simplify the Bookean expression . . . . . .a b c a b c a b cc c c c c c� � using Boolean algebra identities. (N/D 2012)

23. In any Boolean algebra, show that � � � � � � � � � � � �a b b c c a a b b c c ac c c c c c� � � � � � .

(N/D 2013),(M/J 2014)

24. Show that a lattice homomorphism on a Boolean algebra which preserves 0 and 1 is a Boolean homomorphism. (N/D 2013)

25. Let B be a finite Boolean algebra and let A be the set of all atoms ofB . Then prove that the Boolean algebra B is isomorphic to the Boolean algebra ( )P A , where ( )P A is

the power set of A . (M/J 2012)


26. Prove that 110D , the set of all positive divisors of a positive integer 110, is a Boolean

algebra and find all its sub algebras. (A/M 2011)

----All the Best----


MA6566 - Discrete Mathematics -...

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