Unit i) 1)verify that [p->(q->r)] -> [(p->q)->(p->r)] is a tautology or not? 2) simplify the Boolean expression xy+(x+y) + y

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1 Unit i) 1)verify that [p->(q->r)] -> [(p->q)->(p->r)] is a tautology or not? 2) simplify the Boolean expression xy+(x+y) + y 3)find a minimal sum of product representation f(w,x,y)= m(1,2,5,6) 4)simplify the expression 5) Negate and simplify x [p(x) V q (x) ] Unit II) 6) if n is +ve integer prove that n(n+1)= n(n+1)(n+2) / 3 using mathematical induction 7) determine the sequence generated by the generating function f(x) / (1-x) 8) let A={1,2,3,4} B={1,2,3,4,5,6} How many functions are there from f: A-> B How many functions are there from A-> B are one to one How many functions are there from B-> A? how many of these are onto? 9)State and prove Euclidean algorithm? 10)s.t if 8 people are in a room at least 2 of them have birthdays that occur on the same day of the week. Unit iii) 11) In how many ways 5 number of a s 4 number of b s and 3 number of c s can be arranged so that all the identical letters are not in a single block? 12) ) How many derangements are there for 1,2,3,4,5? 13) determine the number of +ve integers n where 1 n 2000 that are i) not divisible by 2, 3, or 5 ii) not divisible by 2,3,5 or 7 iii) not divisible by 2,3,5 but are divisible by 7 14) find how many integer solutions there are to x 1+ x 2+ x 3 + x 4 = 19 if i) 0 x i ii)

2 15) in how many ways one can arrange the letters in the word INFORMATION so that no pair of consecutive letters occurs more than once? Unit IV) 15) Solve the recurrence relation = 0 where n 0 and a 0 =4, a1 =13 16) Write short notes on divide and conquer algorithm. explain binary search examples? 17)define the terms Homomorphism, Isomorphism? 18) Find the general solution for the recurrence relation 4 a n 5 a n = 0, n 1 Unit V ) 19) define the terms complete graph, planar graph, regular graph, walk, path, circuit, cycle, spanning tree, sub graph 20) How many sub graph H=(V,E) of k6 satisfy v = 4? 21)if a tree has 4 vertices of degree 2, one vertex of degree 3, two of degree 4 and one of degree 5 how many pendent vertices does it have? 22) What is minimal spanning tree? Explain the kruskals algorithm with example? 23) prove that every subgroup of a cyclic group is cyclic.

3 CHAITANYA BHARATHI INSTITUTE OF TECHNOLOGY BRANCH: M.C.A ANNUAL EXAMINATION year/sem : I/I SUBJECT: DISCRETE MATHEMATICS(CBCA 101) TIME: 3 HRS TOTAL MARKS: Answer any 5 questions from 5X15=75 1 Answer all questions 5X3=15 a) Construct the truth table for [(p->q)^(q->r)]->(p->r) and verify it is a tautology or not? b) let A={1,2,3,4} and B={1,2,3,4,5,6} How many functions are there from A to B? How many of these are One-to-One? c) Find the coefficient of X 5 in (1-2x) -7 d) Write short notes on Divide And Conquer algorithm? e) State and prove EULER S formula 2 a) Find the minimal product of sums of a function g(w,x,y,z)= M(1,5,7,9,10,13,14,15) Using k-map b) Simplify the expression (AUB) C U B 7M 3 a) show that if 8 people are in a room, at least 2 of them have birthdays that occur on the same day of the week. 7M b) let A={1,2,3,4,5} X (1,2,3,4,5} and define R on A by (x 1,y 1) R (x 2,y 2 ) if x1+y1= x 2 + y 2, verify that R is an equivalence relation on A and also find the equivalence classes [(1,3)],[(2,4)] and [(1,11)] 4 a) find the number of integer solutions of the equation x 1 + x 2 + x 3 + x 4 =18 under the condition X 7 And for all 1 i 4 b) Find the generating function for the sequence 0 2,1 2,2 2, 3 2, 7M

4 5 a) solve the recurrence relation a n - 6 a n a n-2 = 0, n 2,a 0 = 5,a 1 =12 b) Show that Group(G,*) for every a,b є G, (ab) 2 =a 2 b 2 iff (G,*) is an abelian 7M 6 a) Explain in detail about Depth first search (D.F.S) algorithm? 7M b) Explain about kruskal s algorithm for finding minimal spanning trees? 7 Answer any 3 questions a) Define both conjunctive and disjunctive normal forms (c.n.f and d.n.f)? b) Define lattice? Give example c) Define Derangement? How many Derangements are there for 1,2,3,4? d) Define Homomorphism and Isomorphism? e) List out tree traversal techniques CHAITANYA BHARATHI INSTITUTE OF TECHNOLOGY BRANCH: M.C.A ANNUAL EXAMINATION year/sem : I/I SUBJECT: DISCRETE MATHEMATICS(CBCA 101) TIME: 3 HRS TOTAL MARKS: Answer all questions from part A. 5X5=25 Answer 5 questions from part B. 5X10=50 PART-A 1) a) Construct the truth table for [(p->q)^(q->r)]->(p->r) and verify it is a tautology or not? b) Define both conjunctive and disjunctive normal forms (c.n.f and d.n.f)? 2) a) let A={1,2,3,4} and B={1,2,3,4,5,6} How many functions are there from A to B? How many of these are One-to-One? b) Define lattice? 3) a) Find the coefficient of X 5 in (1-2x) -7 b) Define Derangement? How many Derangements are there for 1,2,3,4?

5 4) a) Write short notes on Divide And Conquer algorithm? b) Define Homomorphism and Isomorphism? 5) a) State and prove EULER S formula b)list out tree traversal techniques PART-B 1 a) Find the minimal product of sums of a function g(w,x,y,z)= M(1,5,7,9,10,13,14,15) Using k-map b) Simplify the expression (AUB) C U B 2 a) show that if 8 people are in a room, at least 2 of them have birthdays that occur on the same day of the week. b) let A={1,2,3,4,5} X (1,2,3,4,5} and define R on A by (x 1,y 1) R (x 2,y 2 ) if x1+y1= x 2 + y 2, verify that R is an equivalence relation on A and also find the equivalence classes [(1,3)],[(2,4)] and [(1,11)] 3 a) find the number of integer solutions of the equation x 1 + x 2 + x 3 + x 4 =18 under the condition X 7 And for all 1 i 4 b) Find the generating function for the sequence 0 2,1 2,2 2, 3 2, 4 a) solve the recurrence relation a n - 6 a n a n-2 = 0, n 2,a 0 = 5,a 1 =12 b) prove that Every subgroup of a cyclic group is cyclic 5 a) Explain in detail about Depth first search (D.F.S) algorithm? b) Explain about kruskal s algorithm for finding minimal spanning trees? 6 a) p (q->r) q p validate this argument P

6 ... P b) find a formula to express n 2 as a function of n. 7) a) State and prove Euclidian algorithm b) Show that Group(G,*) for every a,b є G, (ab) 2 =a 2 b 2 iff (G,*) is an abelian CHAITANYA BHARATHI INSTITUTE OF TECHNOLOGY BRANCH: M.C.A ANNUAL EXAMINATION year/sem : I/I SUBJECT: DISCRETE MATHEMATICS(CBCA 101) TIME: 3 HRS TOTAL MARKS: Answer all questions from part A. 5X5=25 Answer 5 questions from part B. 5X10=50 PART-A 1) a) Construct the truth table for [(p->q)^(q->r)]->(p->r) and verify it is a tautology or not? b) Define both conjunctive and disjunctive normal forms (c.n.f and d.n.f)? 2) a) let A={1,2,3,4} and B={1,2,3,4,5,6} How many functions are there from A to B? How many of these are One-to-One? b) Define lattice? 3) a) Find the coefficient of X 5 in (1-2x) -7 b) Define Derangement? How many Derangements are there for 1,2,3,4? 4) a) Write short notes on Divide And Conquer algorithm? b) Define Homomorphism and Isomorphism? 5) a) State and prove EULER S formula b)list out tree traversal techniques

7 PART-B 1 a) Find the minimal product of sums of a function g(w,x,y,z)= M(1,5,7,9,10,13,14,15) Using k-map b) Simplify the expression (AUB) C U B 2 a) show that if 8 people are in a room, at least 2 of them have birthdays that occur on the same day of the week. b) let A={1,2,3,4,5} X (1,2,3,4,5} and define R on A by (x 1,y 1) R (x 2,y 2 ) if x1+y1= x 2 + y 2, verify that R is an equivalence relation on A and also find the equivalence classes [(1,3)],[(2,4)] and [(1,11)] 3 a) find the number of integer solutions of the equation x 1 + x 2 + x 3 + x 4 =18 under the condition X 7 And for all 1 i 4 b) Find the generating function for the sequence 0 2,1 2,2 2, 3 2, 4 a) solve the recurrence relation a n - 6 a n a n-2 = 0, n 2,a 0 = 5,a 1 =12 b) prove that Every subgroup of a cyclic group is cyclic 5 a) Explain in detail about Depth first search (D.F.S) algorithm? b) Explain about kruskal s algorithm for finding minimal spanning trees? 6 a) p (q->r) q p validate this argument P P b) find a formula to express n 2 as a function of n. 7) a) State and prove Euclidian algorithm b) Show that Group(G,*) for every a,b є G, (ab) 2 =a 2 b 2 iff (G,*) is an abelian

NORTHERN INDIA ENGINEERING COLLEGE, LKO D E P A R T M E N T O F M A T H E M A T I C S. B.TECH IIIrd SEMESTER QUESTION BANK ACADEMIC SESSION

NORTHERN INDIA ENGINEERING COLLEGE, LKO D E P A R T M E N T O F M A T H E M A T I C S B.TECH IIIrd SEMESTER QUESTION BANK ACADEMIC SESSION 011-1 DISCRETE MATHEMATICS (EOE 038) 1. UNIT I (SET, RELATION,