SRI VENKATESWARA COLLEGE OF ENGINEERING AND TECHNOLOGY MA DISCRETE MATHEMATICS

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1 1 MA DISCRETE MATHEMATICS UNIT I - LOGIC AND PROOFS Propositional Logic Propositional equivalences-predicates and quantifiers-nested Quantifiers- Rules of inference-introduction to Proofs-Proof Methods and strategy PART A 1. Define conjunction and draw a truth table for it. 2. Define disjunction and draw a truth table for it. 3. Define disjunction and draw a truth table for it. 4. Give the truth table for conditional statement. 5. Write the truth table for bi-conditional statement. 6. Define a tautology. 7. Define a contradiction. 8. Define functionally complete set of connectives 9. State De Morgan laws. 10. Negate and simplify the statement ( p q) r 11. State the truth value of If tigers have wings then the earth travels round the sun. 12. Write the negation of the following preposition: To enter into the country you need a passport or a voter registration card. 13. Construct the truth table for q (p q) p 14. Whether (p q ) ( p q) is a tautology 15. S.T p (p q) is a tautology. 16. Verify (p q) p is a tautology. 17. Define conjunctive normal form. 18. Write the formula without conditional statement ( p q) r 19. Write the formula free from conditional statement p ( q r ) 20. Write down the implication for modus ponens, modus tollens, hypothetical syllogism and dilemma 21. Give the converse and contra positive of the implication If it is raining then I get wet. 22. P.T ( p q ) q p 23. S.T the set [, ] is not a functionally complete set of connectives. 24. Using the truth table show that p (q r) (p q) r 25. S.T the statement is a contradiction (p q) ( p q) ( use Table) 26. Explain the two types of quantifiers with example. 27. Symbolize: Given a positive integer, there is a greatest positive integer 28. Symbolize: For every x, there exist a t s.t x 2 + y Using the predicate calculus symbolize x is the brother of the sister of y 30. Symbolize the expression All the world loves a lover 31. Express 2 is an irrational number using quantifiers. 32. Symbolic form Everyone who is healthy can do all kind of work 33. Symbolic form : The sum of two positive integer is positive 34. Symbolic form: (a) All babies are innocent (b)there is an integer such that it is odd and prime. 35. Using quantifiers, express the following statements over the universe, the set of rational numbers (a) 5 is irrational (b) Subtraction of two rational numbers is rational

2 2 PART B Construct the truth table 1) ( p ( q r) ) (q r ) ( p r ) Show the following 2) q (p q) ( p q) is a tautology. 3) ((p q) ~(~p ( ~ q ~ r))) (~p ~q) ( ~p ~r) is a tautology 4) Test the formula for tautology or contradiction q (p q ) ( p q ) 5) (p q) (r q) (p r) q. 6) S.T ( p q) (r q) (p r) q 7) ( ~ p ( ~ q r)) (q r) (p r) r 8) ( p (q s) ( r p) q r s 9) ( p q ) ( p ( p q )) ( p q) 10) S R is tautologically implied by ( p q ) ( p r ) ( q s) Find the PDNF & PCNF of the following 11) ( a c) ( b a) 12) p (( p q) ~(~q ~ p) 13) (p ( q r )) ( ~ p ( ~q ~r )) 14) p ( ~ p ( q ( ~ q r))) S.T the set of premises are inconsistent 15) p q, q s, r ~ s and p r 16) r (p q) from p q, q r, p m & ~m. 17) Derive: p s from ~ p q, ~q r, r s 18) Prove by indirect method: p q, q r, p v r r 19) Prove by rule cp: derive p ( q s) from : p ( q r), q ( r s) Discuss the validity of the argument 20) a)all educated persons are well behaved. b) Ram is educated. c)no well behaved person is quarrelsome, d) Therefore Ram is not quarrelsome. 21) a)if Abraham works hard then either Bob or Caroline will enjoy themselves. b) If Bob enjoys himself then Abraham will not work hard. c)if Daryl enjoys himself the Caroline will not. S.T If Abraham works hard, then Daryl will not enjoy himself d) It is not sunny this afternoon and it is colder than yesterday e) we will go swimming only if it is sunny f) If we do not go swimming then we will take a canoe trip and g) if we take a canoe trip, then we will be home by sunset lead to the conclusion we will be home by sunset 22) a)if Jack misses many classes through illness, then he fails high school b) If Jack fails high school, then he is uneducated

3 3 c)if Jack reads a lot of books then he is not uneducated d) Jack misses many classes through illness and reads a lot of books. 23) a)if the contract is valid then John is liable for penalty b) If John is liable for penalty, he will go bankrupt. c)if the bank will loan him money, he will not go bankrupt. As a matter of fact, the contract is valid and the bank will loan him money. 24) Use the indirect method of proof in establishing the implication ( x)[(p(x) Q(x)] ( x)[p(x)] ( x)[ Q(x)] 25) Prove formally : ( x)[ P(x)] ( x)[ (P(x) Q(x)] Using a counter example S.T the converse is not true 26) Let A= { 1, 2, 3, 4, 5, 6 } determine the truth value of each of the following (i) ( x A) { x 2 > 25} (ii) ( x A) { x + 6 > 12 (iii) ( x A) { x 2 x < 30 ) 27) S.T from (a) ( x)[(f(x) S(x)] ( y)[ M(y)] W(y) (b) ( y)[(m(y) ~ W(y)] the conclusion ( x)[ F (x) ~ S(x) follows. Verify the validity of the following arguments 28) a)some students attend logic lectures diligently. b) No student attends boring logic lecture diligently. c)sean s lectures on logic are attended diligently by all students. d) Therefore none of Sean s logic lectures are boring. 29) a)every living things is a plant or an animal. b) David s dog is alive and it is not a plant. c)all animals have hearts. d) Hence David s dog has a heart. 30) a)if an integer is divisible by 10 then it is divisible by 2. b) If an integer is divisible by 2 then it is divisible by 3. c)therefore an integer divisible by 10 is divisible by 3

4 4 UNIT IV ALGEBRAIC STRUCTURES Algebraic systems-semi groups and monoids-groups-subgroups and homomorphisms-cosets and Lagrange s theorem- Ring & Fields (Definitions and examples) Ref: Discrete Mathematical Structures with Applications to Computer Science Tremblay & Manohar Sec 3.1, 3.2, 3.5 PART A 1. Define algebraic system with an example. 2. Define homomorphism with example. 3. Define epimorphism with example 4. Define monomorphism with example 5. Define isomorphism with example. 6. Define endomorphism with example 7. Define automorphism with example 8. Define sub algebra with example. 9. Define factor algebra with example. 10. Define semi group with example. 11. Define monoid with example. 12. Define semi group homomorphism. 13. Define monoid homomorphism 14. Define sub semi group with example. 15. Define sub monoid with example. 16. Define direct product with example. 17. Give an example which is a semi group but not monoid P.T the identity element of a group is unique. 20. P.T: Intersection of two normal subgroup of a group is a normal subgroup. 21. Give a necessary and sufficient condition for non-empty subset of a group( G *) to be a subgroup of G 22. S.T by using an example the union of two subgroup of a group G need not be a subgroup 23. Let G be a group s.t x 2 = x for all x (G. S.T G is abelian. 24. Find the GLB & LUB of the set {3, 6, 12 } if they exist in the poset ( Z+, / ) where a/b stands for a divides b. 25. Give an example for non- abelian group. 26. Define normal sub group 27. Define group codes. 28. Give an example of a finite non abelian group. PART B 1) S.T if every element in a group is its own inverse, then the group must be an abelian. 2) S.T every finite group of order n is isomorphic to a permutation group of degree n. 3) If f: G G1 is a group homomorphism P.T Ker f is a normal subgroup of G. 4) Prove that the collection of even permutations is a normal subgroup of the symmetric group.

5 5 5) P.T the intersection of any two normal subgroup of a group is normal. 6) Define a normal subgroup of a group. Give an example. Let G be a group and N ={x G: xy = yx, y G} S.T N is a normal sub group of G. 7) Define a coset of a subgroup H of a group G. Find all the left cosets of the subgroup H = ( p1, p2) of the symmetric group S3 where_ EMBED Equation.3 _Is H a normal subgroup of G? 8) Define the term weight and distance in the context of group codes. S.T the minimum weight of the nonzero code words in a group code is equal to its minimum distance. 9) S.T the set N of natural number is a semi group under the operation x * y = max{x, y}. Is it a monoid? Is the operation commutative? 10) State & prove Lagrange s theorem for groups. 11) Let (G,*) and (H,*) be groups and g: G ( H be a homomorphism. Then prove that the kernel of g is a normal subgroup of G. 12) P.T every subgroup of a cyclic group is cyclic. 13) S.T a non empty subset of H of a group ( G,* ) is a subgroup of G iff a,b H ( a * b -1 H 14) P.T every cyclic group of order n isomorphic to the group (Z n, t n ). P.T the order or a group of a finite group divides the order of the group. 15) P.T the subgroup of a cyclic group is also cyclic 16) State and prove the cayleys theorem on permutation group 17) S.T every subgroup of a cyclic group is normal 18) Define subgroup If A & B are two subgroup of (G,*), P.T A B is a subgroup of (G,*). Is A U B a subgroup of (G,*) 19) P.T any semigroup can be extended to monoid by adjoining an identity element. P.T for any commutative monoid ( M,*) the set of idempotent elements of M forms a submonoid UNIT V LATTICES AND BOOLEAN ALGEBRA Partial ordering-posets-lattices as Posets- Properties of lattices-lattices as Algebraic systems Sub lattices direct product and Homomorphism-Some Special lattices-boolean Algebra PART A 1. Define the transitive closure of a binary relation R with an example 2. Define least and greatest member of a poset. 3. Define a partial ordering relation with an example 4. Let R 1 & R 2 be two compatible relations on a set A. Is R 1 R 2 a compatible relation? 5. If A = { x / x 2 1 = 0} and B = { x / x 2-3x + 2 = 0 }Find A B and A B 6. Define partial order relation and find a partial ordering on the power set of a given set. 7. Let T = {1, 2, 3, 4, 5}. How many subset of T have less than 4 elements? 8. Give an example of a relation on a set which is neither symmetric nor anti symmetric 9. Draw a Hasse diagram of the set of partitions of Define reflexive closure of a set. 11. Give an example for the composition of two function which is not commutative 12. Define partial order relation. 13. Obtain the power set of { a, b,c } 14. Establish the binary relation on the set A = { 1, 3, 5 }which is a symmetric relation 15. What are disjoint sets? Give an examples 16. List all the proper subsets of { 1,2,3}

6 6 17. Let A = { 1,2,3,4} and r be the relation ( on A. Draw the Hasse diagram of r. 18. Let R be the relation on A = { 2,3,4,6,9} defined by x is relatively prime to y Write R as the set of order pairs. 19. Define least element of a poset. 20. Define distributive lattice. 21. In a lattice ( L, ) using usual notation S.T a ( a b ) = a for all a,b L 22. Define a lattice 23. Simplify the Boolean function (f(x, y) = x+ xy 24. In a Boolean algebra, P.T a = b ( ab + a b = In a Boolean algebra, S.T ( a+ b) ( a + c)= a b + ac + bc 26. In a Boolean Algebra a ( a b)= a b 27. P.T every finite lattice is bounded. 28. In Boolean algebra B, P.T a + a.b = a where a,b B. 29. List The three elements sub sets of { a,e.i,o,u} 30. Is ( z, - ) is a semi group. Justify. 31. P.T in a lattice (L, ), a 0 = 0 a = a L. 32. If A= {2,3} ( X = { 2,3,6,12,24,36} and the relation is s.t x y if x divides y, find the least and greatest element for A 33. Draw the Hasse diagram of (X, ) where X is the set of positive divisors of 45 and the relation is such that = { (x,y): x A, y A (x divides y)} 34. Is the lattice of devisors of 32 a Boolean algebra? PART B 1) Let (P, ) be a partially ordered set. Suppose the length of the largest chain in P is n. Then show that the elements in P can be partitioned into a disjoint antichain. 2) Let R be a binary relation on the set of all strings of 0 s & 1 s such that R= {(a,b) / a & b are strings that have the same number of 0 s }. Is R reflexive? Symmetric? Anti symmetric? Transitive? An equivalence relation? A Partial order relation 3) If A,B and C are sets prove that A (B C) = ( A B ) (A C) 4) Let X = ({ 1, 2, 3, 4, 20} and R ={(x,y) / x-y is divisible by 5 } be a relation on X. S.T R is an equivalence relation 5) Explain WARSHALL s algorithm 6) If A, B and C are any three sets, P.T A- ( B C ) = ( A B ) (A C ). 7) Let X = { 1, 2, 3, 4, 5, 6, 7 } and R = { < x, y > x-y is divisible by 3}. S.T R is an equivalence relation. 8) If A, B & C are sets P.T A ( B C) = ( A B) ( A C) with out Venn diagram. 9) If the relation R is defined by xry iff x= (y (mod 5) on S = ( 1, 2, 3,.20) verify that R is an equivalence relation. Find also the partition of S induced by R. 10) Check whether ( P(S), ) is a poset where P(S) is the power set of S and the relation is subset. If it is a poset, determine whether it is lattice or not. 11) Draw the Hasse diagram for divisibility on the set {1, 2, 3, 5, 11, &13} Is it a lattice? 12) Let R be a binary relation and S= {(a,b)/(a,c) (R and (c,b) R for some c} P.T if R is an equivalence relation, then S is also an equivalence. 13) Given R be a reflexive relation on a set A, S.T R is an equivalence relation iff (a,b) & (a,c) are in R implies that (b,c) is in R.

7 7 14) If R is an equivalence relation on an arbitrary set, P.T the set of all equivalence classes constitute a partition of A. 15) Consider the subset A= {1,7,8} B={ 1,6,9,10} & C = ( 1,9,10} where U = ( 1,2,3,..10}List the non empty mini sets generated by A,B,& C. Do they form a partition of U. 16) If A,B & C are sets. P.T A x ( B C ) = ( A x B ) ( A x C) 17) For any two sets A & B, P.T A ( A B ) = A B 18) Given S = { 1,2,3 10 } and the relation R on S where R = { (x,y) / x+ y = 10 }what are the properties of the relation. 19) Let X = { 1,2,. 7} and R { (x,y) / x- y is divisible by 3 } S.T R is an equivalence relation Draw the graph of R. 20) If the relation R defined on S = {1,2,3, 20} br xry iff x y (mod 5) verify that R is an equivalence relation. Find also partition of S induced by R 21) Simplify the Boolean function f( A, B, C, D) = ( (0, 2, 3, 4, 8, 9, 10, 13) 22) Define a lattice homomorphism. S.T a lattice homomorphism preserves the partial order structure. Give an example for isomorphic lattices. 23) Define a distributive lattices with an example. Let ( L,+, ) be a distributive lattice. S.T for any a, b, c L a*b = a*c and a b = a c imply b = c. Hence S.T If L has a complement, it is unique. 24) S.T any finite Boolean algebra is isomorphic to the Boolean algebra of a power set. 25) For any Boolean algebra, S.T a = b iff ab + a b = 0 26) Define an algebraic lattice L. Define a relation on L in such a way that L become a lattice as a partial ordered set 27) Let R be the set of real numbers 0n [0, 1] and be the usual operation of less than or equal to on R. S.T (R., ) is a lattice. What are the operations to meet and join on this lattice? 28) In a Boolean algebra, P.T(a+b )b + c )(c+a )=(a +b)(b +c)(c +a). 29) In any Boolean algebra S.T (a.b )+(b.a )=(a+b). (a +b ) 30) Define a complemented lattice. P.T the complement a of any element a in a complemented lattice is unique. 31) Let f, f1 & f2 be three functions from B n to B P.T is s(f) = s(f 1 )(s (f 2 ), then f(b) = f 1 (b)(f 2 (b), b B. Hence P.T any function f : B n B is produced by Boolean function. 32) Simplify the Boolean function F(x,y,z) = (x (z)((x (y) ((x(y (z) ((y(z) 33) Find the sum-of-product form of the Boolean function f(x,y,z,w) = xy + yw z 34) Let R be a relation on a set A. Then define R -1 ={(a,b)(axa /(b,a) R. P.T if ( A, R ) is a poset then (A, R -1 ) is also poset. 35) In the Boolean algebra of all divisors of 70, find all subalgebras. 36) Draw the Hasse diagram of (P(A),() where P(A) denotes the power set of A= { a,b,c} 37) Define a Boolean algebra and S.T by an example no five element Boolean Algebra can exist