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VALLIAMMAI ENGINEERING COLLEGE DEPARTMENT OF MATHEMATICS SUB CODE/ TITLE: MA6566 – DISCRETE MATHEMATICS QUESTION BANK Academic Year : 2015 - 2016 UNIT – I LOGIC AND PROOFS PART-A 1. Write the negation of the following proposition. “To enter into the country You need a passport or a voter registration card”. 2. How can this English sentence be translated into a logical Expression? “You can access the Internet from campus only if you are computer Science major or you are not freshman”. 3. Makes a truth table for the statement (p q)∨ (~ p) 4. State the truth table of “If tigers have wings then the earth travels round the sun”. 5. Construct the truth table for (i)P ∧ (P Q) , Ҟ(ii)ሺ ՜ ݍሻ՜ሺ ݍ՜ ሻ 6. Using truth table, show that the proposition p ∨ ¬( p q) is a tautology. 7. When a set of formulae is consistent and inconsistent? 8. What is tautology? Give an example 9. State the rules inference for statement calculus. 10. Construct the truth table for (a) ¬( ¬P ¬Q) (b) ¬( ¬P ¬Q) 11. Show that ሺ ՜ ሺ ݍ՜ ݎሻሻ ՜ ሺሺ ՜ ݍሻ ՜ ሺ ՜ ݎሻሻ is a Tautology 12. Write the Scope of the quantifiers in the formula ݔ൫ሺݔሻ՜ ݕݕ,ݔሻ൯. 13. Find the converse and the contra positive of the implication “If it is a raining the I get wet”. 14. Define contrapositive of a statement. 15. Write the duality law of logical expression? Give the dual of שҏҏ FሻרሺQשTሻ. 16. Using truth table verify that the proposition (P Q)∧ ¬(P Q) is a contradiction. 17. Prove by truth tables that ¬(P Q)⇔ ( ¬P ¬Q)∧ (P Q) 18. Define the term logically equivalent. S.T ሺ ՜ ݎ ݍ՜ ݎand p q ՜ ݎare logically equivalent. 19. Let P(x,y) denote the statement x = y+3. What are the truth values of the Proposition P(1,2) , P(3,0). 20. Write the negation of the statement i)(∃x)(∀y) p( x, y) ii)ݔ ݔ ݔݔ ݔ2ሻ? PART-B 1. a. Give the converse and the contrapositive of the implication “ If it is raining then I get wet. b. Show that ( P ( ) ( ) ( ) q r q r p r r ¬ ∧¬ 2. a. Prove that (P ) ( ( ) q p p q p q ¬ →¬ ∨¬ ⇔¬ b. Find the PDNF and PCNF of the formula ( ( ( ))) P P Q Q R ∨¬ ∨¬ 3. a. Without using the truth tables, find the PCNF of ( ( )) ( ( ) p q r p q r ∧¬ →¬ ∧¬ b. Find the principal disjunctive and conjunctive normal form of the formula (( ) P) ( ) S Q R Q R P ¬ ∨¬ →¬ . 4. a. Find the principal disjunctive normal form (PDNF), if possible ( ) ( ) ( ) p q p q q r ∨¬ www.studentsfocus.com
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VALLIAMMAI ENGINEERING COLLEGE DEPARTMENT OF MATHEMATICS

SUB CODE/ TITLE: MA6566 – DISCRETE MATHEMATICS QUESTION BANK Academic Year : 2015 - 2016 UNIT – I LOGIC AND PROOFS

PART-A

1. Write the negation of the following proposition. “To enter into the country You need a passport or a voter registration card”. 2. How can this English sentence be translated into a logical

Expression? “You can access the Internet from campus only if you are computer Science major or you are not freshman”.

3. Makes a truth table for the statement (p ∧ q)∨ (~ p) 4. State the truth table of “If tigers have wings then the earth travels round the sun”. 5. Construct the truth table for (i)P ∧ (P ∨ Q) , (ii) 6. Using truth table, show that the proposition p ∨ ¬( p ∧ q) is a tautology. 7. When a set of formulae is consistent and inconsistent?  8. What is tautology? Give an example 9. State the rules inference for statement calculus. 10. Construct the truth table for (a) ¬( ¬P ∨ ¬Q) (b) ¬( ¬P ∧ ¬Q) 11. Show that is a Tautology 12. Write the Scope of the quantifiers in the formula , . 13. Find the converse and the contra positive of the implication “If it is a raining the I get wet”. 14. Define contrapositive of a statement. 15. Write the duality law of logical expression? Give the dual of F Q T . 16. Using truth table verify that the proposition (P ∧Q)∧ ¬(P ∨ Q) is a contradiction. 17. Prove by truth tables that ¬(P ↔Q)⇔ ( ¬P ∨ ¬Q)∧ (P ∨ Q) 18. Define the term logically equivalent. S.T ∧ and p ∨ q are logically equivalent. 19. Let P(x,y) denote the statement x = y+3. What are the truth values of the Proposition P(1,2) , P(3,0). 20. Write the negation of the statement i)(∃x)(∀y) p( x, y) ii) 2 ? PART-B

1. a. Give the converse and the contrapositive of the implication “ If it is raining then I get wet.

b. Show that ( P ( ) ( ) ( )q r q r p r r¬ ∧ ¬ ∧ ∨ ∧ ∨ ∧ ⇔

2. a. Prove that (P ) ( ( )q p p q p q¬ ∧ → ¬ ∨ ¬ ∨ ⇔ ¬ ∨

b. Find the PDNF and PCNF of the formula ( ( ( )))P P Q Q R∨ ¬ → ∨ ¬ →

3. a. Without using the truth tables, find the PCNF of ( ( )) ( ( )p q r p q r→ ∧ ∧ ¬ → ¬ ∧ ¬

b. Find the principal disjunctive and conjunctive normal form of the formula (( ) P) ( )S Q R Q R P⇔ ¬ ∨ ¬ → ¬ ∧ ∧ → .

4. a. Find the principal disjunctive normal form (PDNF), if possible ( ) ( ) ( )p q p q q r∧ ∨ ¬ ∧ ∨ ∧

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b. Find the PCNF if possible ( ) ( )p r p q¬ → ∧ ↔

5. a. Symbolize the statement “Given any positive integer, there is a greater positive integer” (i) With the positive integers as universe of discourse.

(ii) Without positive integers as universe of discourse.

b. Show that ( )R P Q∧ ∨ is a valid conclusion from the premises P Q∨ , ,Q R P M→ →

and M¬

6. a. Show that d can be derived from the premises ( ) ( ), ( ),a b a c b c d a→ ∧ → ¬ ∧ ∨

b. Show that R S∧ can be derived from the premises , , , ( )P Q Q R R P R S→ → ¬ ∨ ∧

7. a. Show that R S∨ is a valid conclusion from the premises , , ( )C D C D H H A B∨ ∨ → ¬ ¬ → ∧ ¬ and ( ) ( )A B R S∧ ¬ → ∨

b. Show J S∧ logically follows from the premises , , , ( )P Q Q R R P J S→ → ¬ ∨ ∧

8. a. Show that premises , , ,R QR S S Q P Q P→ ¬ ∨ → ¬ → are inconsistent

b. Using CP or otherwise obtain the following implication.

( ( ) ( ); ( ( ) ( )) ( ( ) ( )x p x Q x x R x Q x x R x P x∀ → ∀ → ¬ ⇒ ∀ → ¬

9.a. Show that the following premises are inconsistent:

(i) If Jack misses many classes through illness, then he fails high school. (ii) If Jack fails high school, then he is uneducated (iii) If Jack reads a lot of books then he is not uneducated

(iv) Jack misses many classes through illness and reads a lot of book

b. Prove that ( )xM x∃ follows logically from the premises ( ( ) ( ))x H x M x∀ → and ( )xH x∃

10. a. Prove that ( ( ) ( )) ( ) ( )x P x Q x xp x xQ x∃ ∧ ⇒ ∃ ∧ ∃ . Is the converse true?

b. 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”.

11. a. i) By indirect method, prove that ( ( ) ( ), ( ) ( )x p x Q x xP x xQ x∀ → ∃ ⇒ ∃

ii) Prove that √2 is irrational by giving a proof by contradiction.

b. Show that the premises “ One student in this class knows how to write programs in JAVA” and “Everyone who knows how to write programs in JAVA can get high paying job” imply the conclusion “Someone in this class can get a high paying job”.

12. a. Let p, q and r be the following statement:

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p: I will study discrete mathematics. q: I will watch T.V. r: I am in a good mood. Write the following statements in terms of p, q, r and logical connectives. A). If I do not study discrete mathematics and I watch T.V., then I am in a good mood B) If I am in a good mood, then I will study discrete mathematics or I will watch T.V. C) If I am not in a good mood, then I will not watch T.V. or I will study discrete Mathematics. D) I will watch T.V. and I will not study discrete mathematics if and only if I am in a good mood.

b. Prove that (i) ( ( ) ( ), ( ( ) ( ) ( ( ) ( )x P x S x x P x R x x R x S x∃ ∧ ∀ → ⇒ ∃ ∧

(ii) Without using truth table show that

UNIT – II COMBINATORICS

Part-A

1. What is the number of Permutations of the letters of the word PEPPER? 2. Find the number of Permutations of the letters of the word MATHEMATICS.

2. State Pigeonhole principle. 3. How many positive integers not exceeding 100 that is divisible by 5? 4. What is the minimum number of students required in discrete mathematics class

to be sure that at least six will receive the same grade, if there are five possible

5. Find the recurrence relation for the sequence for 6. Find the number of non-negative integer solutions of the equation , where , , are non-negative integer less than 6. 7. How many words of three different letters can be formed from the letters of the word

MATHEMATICS?

8. Find the recurrence relation for the Fibonacci sequence.

9. Compute the number of distinct 13 card hands that can drawn from a deck of 52 cards.

10.Find the coefficient of x10 in (x+x2+x3+…)3.

11. State the principles of induction.

12. Show that ( 1)1 2 3 . . .2

n nn ++ + + + = by using mathematical induction.

13. using mathematical induction prove that 2 ( 1)nn n< > .

14. If seven colours are used to paint 50 bicycles, then show that at least 8 bicycles will be the same colour.

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15. Find the power set of a finite set 1,2,3 16. Using induction prove that n3+2n is divisible by 3 for all integers n 1. 17. Find the recurrence relation of ( 3) (4) 0.n n

ny A B for n= − + ≥

18. Among 200 people how many of them were born in the same month?

19. Find the homogeneous solution of 1 27 10 6 8n n nS S S n− −− + = + .

20. What is the characteristic equation of the recurrence relation

( ) 2 ( 1) 3 ( 2) 6 ( 3) 0s k s k s k s k+ − − − − − = .

Part-B

1. a. i. Use mathematical induction to prove that 3 7 2n n+ − is divisible by 8 for all 1n ≥

ii. Use mathematical induction to prove that 1 1 1 1........

1.2 2.3 3.4 ( 1) 1n

n n n+ + + + =

+ + for all 1n ≥

b.i. Use mathematical induction to prove that 2 2 2 2 (2 1)(2 1)1 3 5 ....... (2 1)3

n n nn − ++ + + + − = for all 1n ≥

ii. Prove by mathematical induction 6 n+2 + 7 2n+1 is divisible by 43 for each positive integer n.

2. a.i. Prove that 8n – 3n is a multiple of 5 using method of induction

a. ii. Use mathematical induction to prove that 3 2n n+ is divisible by 3 for all integers 1n ≥

b. Find the number of integers between 1 and 250 both inclusive that are

(i) divisible by any of the integers 2, 3. (ii) divisible by 2,3,5.

(iii) not divisible by 2,3,5.

3. a. In a survey of 100 students , if was found that 40 studied mathematics , 64 studied physics 35 studied chemistry , 1 studied all 3 subjects , 25 studied maths and physics , 3 studied math’s and chemistry and 20 studied physics and chemistry . Find the number of students who studied chemistry only.

b. i. How many 1-1 functions are there from a set with m element to a set with n elements?

ii. How many factors has 720?

4. a. i. Find the coefficient of x10 in (1+x5+x10+……….) 3

ii . How many positive integers not exceeding 1000 are divisible by 7 or 11?

b. Find the minimum number of students needed to make sure that 5 of them take the same engineering course ECE, CSE, EEE and Mech.

5. a. the generating function to solve the recurrence relation a n+2 – 8a n+1+ 15a n = 0

given that a0 = 2 , a1 = 8.

b. Using the generating function , solve the difference equation yn+2 – y n+1 – 6 yn = 0 , y1 =1, y0 =2.

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6. a. From the integers from 1 to 100, both inclusive 4-permutations are taken which contain 3 consecutive integers in increasing order, but not necessarily in consecutive positions. How many such permutations are there?

b. Sole the recurrence relation for the Fibonacci sequence 1,1,2,3,5,8,13,………

7. a. Find the recurrence relation and give initial conditions to find the number of n-bit strings that do not have two consecutive 0’s. How many such 5-bit string are there?

b. Using mathematical induction prove that 1

cos( 1) .sin2 2cos

sin2

n

i

x nxnix x=

+=∑ , where n is a positive integer and 2x rπ≠

.

8. a. Show that n th Fibonacci number 2

1 5 32

n

nF n−

⎛ ⎞+≥ ∀ ≥⎜ ⎟⎜ ⎟

⎝ ⎠

b. How many integers between 100 and 999 inclusive a) are divisible by 7? b) are not divisible by 4? c) are divisible by 3 or 4? d) are divisible by 3 and 4? e) are divisible by 3 but not by 4?

9. a. In how many arrangements of the letters of the word PHOTOGRAPH are there with exactly 5 letters between the two Hs ?

b. Prove that number of derangement of n objects is 1 1 1 1 1! 1 ........ ( 1)1! 2! 3! 4! !

nnD n

n⎡ ⎤= − + − + + −⎢ ⎥−⎣ ⎦

10.a. A friend writes a letters to six friends and places them in addressed envelope. In how many ways can he place the letters in the envelopes so that

a) all the letters in the wrong envelops.

b) at least two of them are in the wrong envelopes

ii. Use mathematical induction to show that n3-n is divisible by 3, for n Z +∈

b. Solve the following recurrence relation a n+2 – 2 a n+1 + an = 2n with a0 = 2 , a1 = 1 using generating functions.

11. a. Solve the recurrence relation ( ) ( 1) ( 2) 0, (0) 1, (1) 1F n F n F n F F− − − − = = = b. Solve S(n)-2S(n-1)-3S(n-2)=0, n ≥ 2 with S(0) = 3 and S(1) = 1 by using generating function.

UNIT – III GRAPH THEORY

1. How many edges are there in a graph with 10 vertices each of degree 5? 2. Define regular graph and a complete graph. 3. What is meant by isomorphism of graphs? 4. Define a complete graph and give an example. 5. Define Pendant vertex in a graph and give an example 6. Define Regular graph and give an example

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7. Define a strongly connected graph and give an example 8. Find number of edges and degree of each vertex in the complete graph K5 9.Does there exists a simple graph with the degree sequence 3 , 3 , 3 , 2 ? 10.Define Euler paths. Give an example 11. Determine whether the graph G in figure has Euler path. Construct such a path if it exists.

12. Define a Hamilton Path in G. Give an example. 13. A regular graph G has 10 edges and degree of any V is 5, find the number of vertices. 14. Define Pseudo graph. Give an example 15. Draw a complete bipartite graph of K2,3 and K3,3. 16. Define isomorphism of two graphs. Give an example 17.State the condition for the multigraph to be traversable? 18.Give an example each for connected and disconnected graphs. 19.Find all cut vertices and cut edges of the graph G given below.

20.Test whether the graph G is Eulerian

PART-B

1. a. Are the simple graph with the following adjacency matrices 0 0 1 0 1 00 0 1 , 1 0 01 1 0 1 0 0

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

isomorphic? b. Define the graph isomorphic and give an example of isomorphic and non-isomorphic graphs.

2. a. Determine whether the graphs G and H are isomorphic

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b. Find all cut vertices and cut edges of the following graph

3. a. Find the adjacency matrix “A” of the following graph

Find 2A and 3A . What are your observations recording the entries in 2A ? b. Write the adjacency matrix of the digraph

G = ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ){ }1 3 1 2 2 4 3 1 2 3 3 4 4 1 4 2 4 3, , , , , , , , , , , , , , , ,v v v v v v v v v v v v v v v v v v

4. a. A connected graph G is Eulerian if and only if every vertex of G is of even degree. b. Prove that if a graph G has not more than two vertices of odd degree, then there can be Euler path in G. 5. a. Show that the K7 has Hamiltonian graph. How many edge disjoint Hamiltonian cycles are there in K7? List all the edge-disjoint Hamiltonian cycles. Is it Eulerian graph? b. Define Eulerian graph and Hamiltonian graph. Give an example of a graph which is Eulerian but not Hamiltonian and vice versa. 6. a. Show that in a simple digraph, every node of the digraph lies in exactly one strong component. b. Draw the graph whose adjacency matrix given below

0 0 1 10 0 1 01 1 0 11 1 1 0

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

7. a. Establish an isomorphism between the graphs G and H G H

C D

B A

HG

FE

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b. Define bipartite graph. Show that if G is bipartite simple graph with p vertices and

q edges then ⎥⎦

⎤⎢⎣

⎡≤

4

2pq

8. a. Define complement of a graph. Find the complement G of the following graph G.Is it true that G is isomorphic to G . Justify your answer b. Show that the complete bipartite graph nnK , has the Hamiltonian cycle. When nnK , has Eulerian circuit? Justify your answer 9. a. Define the degree of a vertex and prove that the number of vertices of odd degree is always even. b. Find the Euler path or an Euler circuit, if it exists, in the following graphs.

10. a. Define a complete graph nk , Draw a complete graph 6k . What is the degree of each vertex in nk ? What is the total number of edges in nk ? b. Define complement of a graph. Find the complement G of the following graph G. Is it true that G is isomorphic to G G

11.a.If G is a simple graph with n vertices and k components, then the number of edges is at most ( )( 1) / 2n k n k− − + b. Prove that a simple graph with n vertices must be connected if it has more than

2)2()1( −− nn edges.

ED C

B A

E D

C

B

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UNIT – IV ALGEBRAIC STRUCTURES

PART-A

1. Define semi group. Give an example.

2. Define monoid. Give an example.

3. Give an example of a non abelian finite group.

4. In an abelian group , , prove that for all a,b G

5. Show that the inverse of an element in a group , is unique

6. Find a subgroup of order two of the group ,

7. State Lagrange’s theorem for finite groups.

8. If H is a subgroup of G, among the right cosets of H in G prove that there is only one subgroup viz.H.

9. Show that the permutation 1 2 3 3 2 5

4 5 61 4 6 is odd.

10. Define ring. Give an example

11. Define subring. Give an example

12. Define field. Give an example.

13. Find an identity element of a group G with binary operation defined by 2

aba b∗ =

14. State and Lagrange’s theorem.

15. If f = 1 2 3 42 3 4 1

⎛ ⎞⎜ ⎟⎝ ⎠

and g = 1 2 3 43 1 4 2

⎛ ⎞⎜ ⎟⎝ ⎠

find 1f gf− .

16. A non-empty subset H of a group (G, ∗ ) is a subgroup of G if and only if

1 ,a b H a b H−∗ ∈ ∀ ∈

17. State and Cayley’s theorem or Cayley’s representation theorem

18. Show that 1 , is an abelian group under the binary operation * defined by

, 1

19. Prove that in a group G the equations a * x = b and y * a = b have unique solutions

for the unknowns x and y as x = a-1 * b and y = b* a-1 where a, b ∈ G.

20. Show that the set of all non-zero real numbers is an abelian group under the operation

∗ defined by 2

aba b∗ =

PART-B

1.a. If ‘S’ is the set of all ordered pairs (a,b) of real numbers with the binary operation ⊕ defined by(a,b) ⊕

(c,d)=(a+c,b+d), where a,b,c,d are real, prove that (S, ⊕ ) is a commutative group.

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b. i. If f = 1 2 3 42 3 4 1

⎛ ⎞⎜ ⎟⎝ ⎠

and g = 1 2 3 43 1 4 2

⎛ ⎞⎜ ⎟⎝ ⎠

find 1f gf− and 1gfg−

ii. ii. If P1 = 1 2 3 4 5 63 2 5 1 4 6 and P2 = 1 2 3 4 5 6

4 2 3 5 1 6

Compute P1 P2. Verify that (P2 P1)-1 = P1-1 P2

-1.

2. a. If f = 1 2 3 4 52 4 5 1 3

⎛ ⎞⎜ ⎟⎝ ⎠

and g = 1 2 3 4 55 2 4 3 1

⎛ ⎞⎜ ⎟⎝ ⎠

are permutation on the set

A = { }1, 2,3, 4,5 , find a permutation g on ‘A’ such that f.g = g.f

b. Prove that the set of all matrices a bb a

⎛ ⎞⎜ ⎟−⎝ ⎠

form an abelian group with respect to matrix multiplication

where a and b are real numbers, not both 0.

3. a. If H1 and H2 are subgroups of a group (G, ∗ ) prove that H1 ∩ H2 is a group of (G, ∗ )

b. Let S be a non-empty set and P(S) denote the power set of S. Verify whether (P(S), ∩ ) is a group. 4. a. Find all non-trivial subgroups of (Z6, +6). b. Determine whether H = { 0, 5, 10} and k = {0, 4, 8, 12} are subgroups of the group

{Z15, +15}.

5. a. State and prove Lagrange’s theorem .

b. If f is a homomorphism from a group (G,∗ ) into ( 'G , .) then prove that

a) f(e) = 'e , where e, 'e are the identities of G and 'G respectively.

b) 1 1( ) ( ( ))f a f a− −= for all a G∈

6. a. Show that Kernal of a group homomorphism is a normal subgroup of the group.

b. State and prove fundamental theorem of group homomorphism

7. a. State and prove Cayley’s theorem or Cayley’s representation theorem

b. Determine all cosets of a subgroup H = {1, a2} of a group G = {1, a, a2, a3} under usual multiplication,

where a4 = 1

8. a. Let G be a group and a ∈ G. Show that the map f : G ÆG defined by f(x) = a x a-1 for every x ∈ G is an

isomorphism.

b. If H is a group of G such that 2x H x G∈ ∀ ∈ , Prove that H is normal subgroup of G.

9. a. Show that monoid homomorphism preserves the property of invertibility.

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b. For any commutative monoid (M,∗ ), Prove that the set of all idempotent elements of

M forms a submonoid.

10. a. Let E = { }2, 4,6,8,..... show that ( , )E + and ( , )E × are semi groups, but not monoids.

b. If H is a subgroup of G, among the right cosets of H in G prove that there is only one

subgroup viz. H

11. a. If f is a homomorphism of a group G into a group G’ then prove that group

Homomorphism preserves identities. `

b. Show that every cyclic group of order n is isomorphic to the group (Zn, +n)

12. a. Prove that intersection of two normal subgroups of a group will be a normal subgroup.

b. Show that if every element in a group is its own inverse, then the group must be abelian.

UNIT – V LATTICES AND BOOLEAN ALGEBRA

PART-A 1. Define a partially ordered set.

2. Define a partial order relation.

3. Draw a Hasse diagram of 1,2,4,5,10,20 .

4. Show that (X,≤) is a chain , where 1,2,3,4,6,12 and ≤ is the usual less than or

equal to relation.

5. If a poset has a least element, then prove that it is unique.

6. Define a Boolean algebra.

7. Give an example of two-element Boolean algebra.

8. Is the lattice of divisors of 32 a Boolean algebra?

9. Prove that . . is a Boolean algebra.

10. Give an example of a lattice which modular but not distributive.

11. Form the table of operations for the Boolean algebra , ,0,1 , given 0,1

12. The following is the Hasse diagram of a partially set. Verify whether it is a lattice.

13. In the following lattice, find

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14. Give an example of a lattice that is not distributive and not modular.

15. Let S = { }, ,a b c then the power set { } { } { } { } { } { } { }{ }( ) , , , , , , , , , , , ,P S a b c a b a c b c a b cφ= is a poset with

respect to the relation inclusion⊆ . Draw the Hasse diagram.

16. Let N be the set of all natural numbers and define m n≤ if n-m is a non negative integers. Show that

( , )N ≤ is a poset.

17. Let N be the set of all natural numbers with the relation R as follows: a R b if and only if a divides b. Show

that R is a partial order relation on N.

18. The following is the Hasse diagram of a partially ordered set. Verify whether it is a lattice.

19. Define a lattice. Verify whether the lattice given by the Hasse diagram in the figure below is distributive.

20. Draw a Hasse diagram of 20D = { }1,2, 4,5,10, 20

PART-B

1.a. If poset has a least element, then prove that it is unique.

b. Let ( , )L ≤ be a lattice in which ∗ and ⊕ denote the operations of meet and join respectively. For any a ,b

∈ L prove that a ≤ b a b a a b b⇒ ∗ = ⇒ ⊕ =

2. a. Let ( , )L ≤ be a lattice. For any a,b,c ∈L the following properties called isotonicity hold.

If b ≤ c then a) a b∗ ≤ a c∗ b) a b a c⊕ ≤ ⊕

c

0

ba

e

d

c ba

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b. Let ( , )L ≤ be a lattice . For any a,b,c ∈L the following inequalities known as distributive inequalities hold.

a) ( ) ( ) ( )a b c a b a c⊕ ∗ ≤ ⊕ ∗ ⊕ b) ( ) ( ) ( )a b c a b a c∗ ⊕ ≥ ∗ ⊕ ∗

3. a. Show that in a lattice ( , )L ≤ if a b≤ and c d≤ , then a) a c b d∗ ≤ ∗ b) a b b d⊕ ≤ ⊕ b.

Prove that every chain is a distributive lattice.

4. a. In a distributive lattice ( , , )L ∗ ⊕ if for any a, b, c ∈ ,L a b a c∗ = ∗ and a b a c⊕ = ⊕

then b = c.

b. State and prove De Morgan’s Laws.

5. a. Prove that every distributive lattice is modular.

b. If 45D denotes the set of all divisors of 45, under divisibility ordering find which elements have

complements and which do not have complements.

6. a. In a lattice prove that a b a b a≤ ⇒ ∗ =

b. In a distributive lattice, show that ( ) ( ) ( ) ( ) ( ) ( )a b b c c a a b b c c a∗ ⊕ ∗ ⊕ ∗ = ⊕ ∗ ⊕ ∗ ⊕

7. a. In a distributive lattice prove that complement of an element, if it exists, is unique

b. In any Boolean algebra, show that

8. a. Form the table of operations for the Boolean algebra ( , , , ',0,1)B ∗ ⊕ , given B = { }0,1

b. If B is a Boolean algebra then prove that for , 1 1a B a∈ + = and .0 0a =

9. a. If x and y are elements in Boolean algebra, then prove that ' 'x y x y≤ ⇒ ≥

b. In a Boolean algebra show that ' ' 0ab a b+ = if and only if a b=

10.a. In any lattice prove that ( ) ( ) ( )a b c a b a c∧ ∪ ≥ ∧ ∪ ∧

b. Show that a lattice homomorphism on a Boolean algebra which preserves 0 & 1

is a Boolean homomorphism.

11. a. Let L be lattice, where a*b = glb(a,b) and a b=lub(a,b) for all a,b . Then both binary operations * &

defined as in L satisfies commutative law, associative law, absorption law and idempotent law.

b. Show that in a distributive and complemented lattice satisfied De Morgan’s law.

12. a. Show that in a distributive and complemented lattice

0 1 . b. Show that in a lattice if , then

.

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