Anna University Coimbatore 2009 B.E Computer Science and Engineering Model , , - Question Paper

DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING

MODEL EXAMINATION

DISCRETE MATHEMATICS

 

PART-A Answer ALL Questions ( 20 X 2 = 40)

1. Define Conditional Statement with Truth table.
2. Construct the truth table for .
3. Show that the proposition is a Tautology.
4. Give the converse and contra positive of the implication If it is raining then I get wet .
5. Obtain the CNF for .
6. Define Simple Statement function with example.
7. Symbolize the expression All the world loves a lover.
8. Show thatfollows logically from the premises .
9. Prove that i) ii) .
10. Find given .
11. Let and . Draw the graph of R and also give its matrix.
12. Let and the relation be such that if x divides y. Draw the Hasse diagram of .
13. Determine whether defined by is one to one and onto.
14. Let and where R is the set of real numbers. Find .
15. Is the permutation even or odd?
16. Find the identity element of the group of integers with the binary operation * defined by .
17. Define Homomorphism.
18. Show that the set N={0,1,2,3} is a semi group under the operation . Is it a monoid?
19. Prove that the semi group homomorphism preserves the property of Commutativity.
20. Determine the parity check code (3,4)

 

PART-B Answer ANY FIVE Questions ( 5 X 12 = 60)

21. i)Construct the truth table for . (6 Marks)

ii) Obtain the PDNF and PCNF of . (6 Marks)

 

 

22. i) Express is an irrational number. (6 Marks)

ii) Show that the conclusion follows from the premises and . (6 Marks)

23. i) Given and a relation R on S where . What are the

properties of the relation R? (6 Marks)

ii) Let A be the given finite set and its power set. Let be the inclusion relation on the

elements of . Draw Hasse diagram of (,) for a) A={a}, b) A={a,b} c)A={a,b,c}

d)A={a,b,c,d} (6 Marks)

24. i)Let where I is the set of integers and . Show that the binary

operation * is Commutative and Associative. Find the identity element and indicate the inverse

of each element. (8 Marks)

ii) If andare permutations. Prove that (4 Marks)

25. i) Show that the group homomorphism preserves identity, inverse and subgroup. (8 Marks)

ii) Prove that the intersection of two normal subgroups is a normal subgroup.(4 Marks)

 

26. i) Show that the following set of premises are inconsistent:

If Rama gets his degree, he will go for a job. If he goes for job, he will get married soon.

If he goes for higher study, he will not get married. Rama gets his degree and goes for

higher study. (6 Marks)

ii) Rewrite the following using quantifiers

a) For every positive integer there is a greater positive integer. b) It is not true that all roads

lead to Rome. c) Some people who trust others are rewarded. (6 Marks)

27. i) In a survey of 100 students, it was found that 40 studied Mathematics, 64 studied Physics, 35 studied Chemistry, 1 studied all the three subjects, 25 studied Mathematics and Physics, 3 studied Mathematics and Chemistry and 20 studied Physics and Chemistry. Find the number of students who studied Chemistry only and the number who studied none of these subjects. (6 marks)

ii) Show that the operation * on Q, the set of rational numbers, defined by is

commutative and associative. What are the identity and inverse elements under *? (6 Marks)

28. i) Let and where R is the set of real numbers given by

Find . State whether these functions

are injective, surjective and bijective (6 Marks)

ii) Let is given by e(000) = 000000, e(001) = 001011 , e(010) = 010101,

e(011) = 011110, e(100) = 100110, e(101) = 101101, e(110) = 110011,

e(111) = 111000. Determine the coset leader for the word received 001110. (6 Marks)