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

Size: px

Start display at page:

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

Ada Osborne
5 years ago
Views:

1 VALLIAMMAI ENGINEERING COLLEGE DEPARTMENT OF MATHEMATICS SUB CODE/ TITLE: MA6566 DISCRETE MATHEMATICS QUESTION BANK Academic Year : 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.. 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 1. 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,), P(3,0). 0. Write the negation of the statement i)( x)( y) p( x, y) ii)? 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. 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 S (( Q R) P) ( Q R) P. 4. a. Find the principal disjunctive normal form (PDNF), if possible ( p q) ( p q) ( q r)

2 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( px ( ) Qx ( ); xrx ( ( ) Qx ( )) xrx ( ( ) Px ( ) 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( Px ( ) Qx ( )) xpx ( ) xqx ( ). 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( px ( ) Qx ( ), xpx ( ) xqx ( ) ii) Prove that 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. 1. a. Let p, q and r be the following statement:

3 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) xpx ( ( ) Sx ( ), xpx ( ( ) Rx ( ) xrx ( ( ) Sx ( ) (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?. Find the number of Permutations of the letters of the word MATHEMATICS.. 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 grades? 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 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 5 cards. 10.Find the coefficient of x 10 in (x+x +x 3 + ) State the principles of induction. 1. Show that nn ( + 1) n = by using mathematical induction. n 13. using mathematical induction prove that n < ( n > 1). 14. If seven colours are used to paint 50 bicycles, then show that at least 8 bicycles will be the same colour.

4 15. Find the power set of a finite set 1,,3 16. Using induction prove that n 3 +n is divisible by 3 for all integers n 1. n n 17. Find the recurrence relation of y = A( 3) + B(4) for n 0. n 18. Among 00 people how many of them were born in the same month? 19. Find the homogeneous solution of Sn 7Sn 1+ 10Sn = 6 + 8n. 0. What is the characteristic equation of the recurrence relation sk ( ) + sk ( 1) 3 sk ( ) 6 sk ( 3) = 0. Part-B n n 1. a. i. Use mathematical induction to prove that is divisible by 8 for all n n ii. Use mathematical induction to prove that = for all nn ( + 1) n+ 1 n n(n 1)(n+ 1) b.i. Use mathematical induction to prove that (n 1) = for all n 1 3 ii. Prove by mathematical induction 6 n+ + 7 n+1 is divisible by 43 for each positive integer n.. a.i. Prove that 8 n 3 n is a multiple of 5 using method of induction a. ii. Use mathematical induction to prove that n 3 + n is divisible by 3 for all integers n 1 b. Find the number of integers between 1 and 50 both inclusive that are (i) divisible by any of the integers, 3. (ii) divisible by,3,5. (iii) not divisible by,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, 5 studied maths and physics, 3 studied math s and chemistry and 0 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 70? 4. a. i. Find the coefficient of x 10 in (1+x 5 +x 10 +.) 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+ 8a n+1+ 15a n = 0 given that a0 =, a1 = 8. b. Using the generating function, solve the difference equation yn+ y n+1 6 yn = 0, y1 =1, y0 =.

5 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,,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 x nx n cos( n + 1).sin cos ix =, where n is a positive integer and x rπ x i= 1 sin. 8. a. Show that n th Fibonacci number Fn 1+ 5 n n 3 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 H s? b. Prove that number of derangement of n objects is Dn n 1 = n! ( 1) 1!! 3! 4! 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 n 3 -n is divisible by 3, for n Z + b. Solve the following recurrence relation a n+ a n+1 + an = n with a0 =, a1 = 1 using generating functions. 11. a. Solve the recurrence relation Fn ( ) Fn ( 1) Fn ( ) = 0, F(0) = 1, F(1) = 1 b. Solve S(n)-S(n-1)-3S(n-)=0, n 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?. 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

7. Define a strongly connected graph and give an example 8. Find number of edges and degree of each vertex in the complete graph K 5 9.

Determine whether the graph G in figure has Euler path. Construct such a path if it exists. 1. Define a Hamilton Path in G. Give an example.

Draw a complete bipartite graph of K,3 and K 3,3. 16. Define isomorphism of two graphs. Give an example 17.

Find all cut vertices and cut edges of the graph G given below. 0.Test whether the graph G is Eulerian PART-B 0 0 1 0 1 0 1. a. Are the simple graph with the following adjacency matrices 0 0 1, 1 0 0 1 1 0 1 0 0 isomorphic?

6 7. Define a strongly connected graph and give an example 8. Find number of edges and degree of each vertex in the complete graph K 5 9.Does there exists a simple graph with the degree sequence 3, 3, 3,? 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. 1. 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 K,3 and K 3, 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. 0.Test whether the graph G is Eulerian PART-B a. Are the simple graph with the following adjacency matrices 0 0 1, isomorphic? b. Define the graph isomorphic and give an example of isomorphic and non-isomorphic graphs.. a. Determine whether the graphs G and H are isomorphic

b. Find all cut vertices and cut edges of the following graph 3. a. Find the adjacency matrix A of the following graph 3 Find A and A.

Write the adjacency matrix of the digraph G = {( v, 1 v3)( v, 1 v ), ( v, v ), 4 ( v, 3 v1), ( v, v3), ( v, 3 v4), ( v, 4 v1), ( v, 4 v), ( v, 4 v 3) } 4. a. A connected graph G is Eulerian if and only if every vertex of G is of even degree.

How many edge disjoint Hamiltonian cycles are there in K 7? List all the edge-disjoint Hamiltonian cycles. Is it Eulerian graph? b. Define Eulerian graph and Hamiltonian graph.

7 b. Find all cut vertices and cut edges of the following graph 3. a. Find the adjacency matrix A of the following graph 3 Find A and A. What are your observations recording the entries in b. Write the adjacency matrix of the digraph G = {( v, 1 v3)( v, 1 v ), ( v, v ), 4 ( v, 3 v1), ( v, v3), ( v, 3 v4), ( v, 4 v1), ( v, 4 v), ( v, 4 v 3) } 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. A? 5. a. Show that the K 7 has Hamiltonian graph. How many edge disjoint Hamiltonian cycles are there in K 7? 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 a. Establish an isomorphism between the graphs G and H A B E F G H D H C G

8 b. Define bipartite graph. Show that if G is bipartite simple graph with p vertices and p q edges then q 4 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 K n, n has the Hamiltonian cycle. When K n, n 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. B A B A C E C D E D 10. a. Define a complete graph k n, Draw a complete graph k 6. What is the degree of each vertex in k n? What is the total number of edges in k n? 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 ( n k)( n k+ 1)/ b. Prove that a simple graph with n vertices must be connected if it has more than ( n 1)( n ) edges.

9 1. Define semi group. Give an example.. Define monoid. Give an example. 3. Give an example of a non abelian finite group. UNIT IV ALGEBRAIC STRUCTURES PART-A 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 Define ring. Give an example 11. Define subring. Give an example 1. Define field. Give an example is odd Find an identity element of a group G with binary operation defined by 14. State and Lagrange s theorem. 15. If f = and g = find 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 f 1 gf a b= 18. Show that 1, is an abelian group under the binary operation * defined by, 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. 0. Show that the set of all non-zero real numbers is an abelian group under the operation ab defined by a 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.. ab

10 1 3 4 b. i. If f = and g = find ii. ii. If P 1 = and P = f 1 gf and gfg 1 Compute P 1 P. Verify that (P P 1 ) -1 = P 1-1 P -1.. a. If f = and g = are permutation on the set A = { 1,, 3, 4, 5 }, find a permutation g on A such that f.g = g.f a b. Prove that the set of all matrices b where a and b are real numbers, not both 0. b form an abelian group with respect to matrix multiplication a 3. a. If H 1 and H are subgroups of a group (G, ) prove that H 1 H 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 (Z 6, + 6 ). b. Determine whether H = { 0, 5, 10} and k = {0, 4, 8, 1} are subgroups of the group {Z 15, + 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) = b) ' e, where e, 1 1 f ( a ) = ( f( a)) for all a G ' e are the identities of G and ' G respectively. 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, a } of a group G = {1, a, a, a 3 } under usual multiplication, where a 4 = 1 8. a. Let G be a group and a G. Show that the map f : G G 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 x H x G, Prove that H is normal subgroup of G. 9. a. Show that monoid homomorphism preserves the property of invertibility.

11 b. For any commutative monoid (M, ), Prove that the set of all idempotent elements of M forms a submonoid. 10. a. Let E = {, 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 (Z n, + n ) 1. 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.. Define a partial order relation. 3. Draw a Hasse diagram of 1,,4,5,10,0. 4. Show that (X, ) is a chain, where 1,,3,4,6,1 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 3 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 1. The following is the Hasse diagram of a partially set. Verify whether it is a lattice. 13. In the following lattice, find

12 14. Give an example of a lattice that is not distributive and not modular. 15. Let S = { abc,, } then the power set PS ( ) { φ, { a},{ b},{ c},{ ab, },{ ac, },{ bc, },{ abc,, }} respect to the relation inclusion. Draw the Hasse diagram. = is a poset with 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. e a c b d 19. Define a lattice. Verify whether the lattice given by the Hasse diagram in the figure below is distributive Draw a Hasse diagram of D 0 = { 1,, 4, 5,10, 0 } a 0 b c 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. 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

13 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 La, b= a cand 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 D45denotes 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 ( B,,,',0,1), given B = { 0,1 } b. If B is a Boolean algebra then prove that for a B, a+ 1= 1 and a.0 = 0 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 ab' + a ' b = 0 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. 1. a. Show that in a distributive and complemented lattice 0 1. b. Show that in a lattice if, then.

MA Discrete Mathematics

MA Discrete Mathematics MA2265 - 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