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SATHYABAMA UNIVERSITY (Established under section 3 of UGC Act,1956) Course & Branch :B.E - P-CSE Title of the Paper :Discrete Mathematics Sub. Code :611PT501 Date :22/04/2010 Max. Marks :80 Time : 3 Hours Session :FN ______________________________________________________________________________________________________________________ PART - A Answer ALL the Questions (10 x 2 = 20) 1. Find the principal disjunctive normal form of (q p) (~ p q). 2. Define tautology and contradiction. 3. Draw the Hasse Diagram for the relation “less than” on the set A = {2, 3, 4, 6, 12, 36, 48}. Here a R b if a less than b for a, b A. 4. If A, B, C are sets such that A B = A C and A B = A C, show that B = C. 5. State Lagrange’s theorem. 6. Find the product and verify the commutative law for the 1 2 3 4 1 2 3 4 f andg 4 3 1 2 permutations 3 1 4 2 7. How many three digit numbers can be formed with the number 5, 7, 9, 1. (a) A digit cannot appear more than once. (b) Any digit may appear any number of times. 8. Define a lattice with an example. 9. Define a complete graph Kn. For what values of n, Kn is regular? 10. Draw a rooted tree for the given expression and also find the value of the expression: + - * 2 3 5 / 234 PART – B Answer All the Questions (5 x 12 = 60) 11. (a) Using the method of direct proof, prove that, r ~ q, r s, s ~ q p q ~ p. (b) Construct the truth table for the given propositions: (p q) (~ p r) (or) 12. (a) Using the rules of predicate calculus, show that (x) (P(x) Q(x)), (x)P(x) (x) Q(x). (b) By using any method of proving, show that R S follows logically from the premises C D, (C D) ~ H, ~ H (A ~ B) and (A ~ B) (R S). 13. (a) Let X = {1, 2, 3,….,25} and R = {(x, y)/ x – y is divisible by 5} be a relation in X. Show that R is an equivalence relation. (b) Let A, B, and C be three sets, f: A B, g: B C,, if f and g are Surjective. Prove that gof is also surjective. (or) 14. (a) Consider the relation R defined on the set {1, 2, 3, 4, 5, 6}and R = {(i, j): | i – j| = 1}. Draw the graph representing the given relation. Check whether it is reflexive, symmetric or transitive. (b) Without using Venn diagram, prove A – (BC) = (A – B) (A – C) 15. (a) Prove that every subgroup of a cyclic group is cyclic. (b) Prove that a group G is abelian iff (a * b)2 = a2 * b2. (or) 16. (a) Prove that a subgroup H of a group G is normal iff x H x-1 = H for all x G. (b) Prove that every cyclic group is abelian. 17. (a) Solve the recurrence relation T(k) – 7T(k – 1) + 10T(k – 2)= 6 + 8k, k 2, with T(0) = 1 and T(1) = 2. (b) Consider the Boolean function. f(x1, x2, x3) = [(x1 x2) x3)] (x1 x2) (i) (ii) (iii) Simplify f algebraically. Draw the switching circuit of f. Draw the same of f obtained in (i). (or) 18. (a) Using mathematical induction, prove that 1. 1! + 2.2! + ……..+ n.n! = (n + 1)! – 1, where ‘n’ is a positive integer. (b) Using the recurrence relation, model the problem and solve. Suppose that a person deposits Rs. 10,000 in a savings account at a bank yielding 11% per year with interest compounded annually. How much will be in the account after 30 years? 19. (a) If exists, find an example for the following. Justify. (i) A weakly connected graph but not strongly connected. (ii) A strongly connected graph but not weakly connected. (iii) Both weakly connected and strongly connected. (iv) Neither weakly nor strongly connected. (b) How many edges are there in a forest of ‘t’ trees containing a total of ‘n’ vertices? (or) 20. (a) Prove that an undirected graph has even number of vertices of odd degree. (b) Prove that a graph G with ‘n’ vertices is a tree iff it has (n – 1) edges.