Reg. No. :

M.Sc. DEGREE EXAMINATION, AUGUST/SEPTEMBER 2017

First Semester

Computer Science

DCS 7105 — MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE

(Regulations 2013)

Time : Three hours Maximum : 100 marks

Answer ALL questions.

PART A — (10 2 = 20 marks)

1. Find the truth table of .~ QP .

2. Negate the statement: 4 is a prime number and 5 is not an even integer.

3. How many words of three distinct letters can be formed from the letters of the

word LAND?

4. Solve the recurrence relation .08621

rrraaa

5. Find the inverse of

132546

654321 , where is a permutation map.

6. Define cyclic group with an example.

7. Simplify the Boolean expression yxx ' .

8. Obtain Hasse Diagram of ,, Ap where dcbaA ,,, .

9. When are two finite state automata said to be equivalent?

10. Draw the state diagram of a FSA that accepts strings over ba , containing

exactly one b.

Question Paper Code : BS2505

BS2505 2

PART B — (5 13 = 65 marks)

11. (a) (i) Obtain the PCNF and PDNF of the formula

.~~~ PQQPP (6)

(ii) Define inconsistent premises. Show that the premises (7)

AEHAHSSE ,~,, are inconsistent.

Or

(b) (i) Show that .~,~,,~ PQPQSSRQR (7)

(ii) By indirect method prove that .,, RSQPSQRP (6)

12. (a) (i) There are 3 piles of identical red, blue and green balls, where each

pile contains at least 10 balls. In how many ways can 10 balls be

selected (1) if there is no restriction? (2) if at least one red ball must

be selected? (3) if at least one red ball, at least 2 blue balls and at

least 3 green balls must he selected? (7)

(ii) Find the number of non-negative integer solutions of the inequality

10654321 xxxxxx . (6)

Or

(b) (i) Prove by mathematical induction, that (7)

3214

121,5.4.34.3.23.2.1 nnnnnnn .

(ii) Solve the recurrence relation .22321

r

rrraaa

(6)

13. (a) (i) Show that two cosets are either identical or disjoint. (6)

(ii) Show that if order of a group is prime, then it has no proper

subgroups. (7)

Or

(b) State and Prove Lagrange’s theorem. (13)

14. (a) (i) Let .30,15,6,5,3,2,1A Show that divides is a partial ordering of A,

and draw the Hasse Diagram. (7)

(ii) In any Boolean algebra, show that 0'' baab if and only if .ba

(6)

Or

(b) (i) If R is the relation on the set of positive integers such that Rba ,

if and only if ba 2 is even, then prove that R is an equivalence

relation. (7)

(ii) Simplify the Boolean expression ',',',',,',' cbacbacba using

Boolean algebra identities. (6)

BS2505 3

15. (a) (i) Construct and an FSA that accepts all strings over ba , which

begin with a and end with b. (6)

(ii) Design an NFA that accepts the non-null strings over ba , starting

with ab but not ending with ab. (7)

Or

(b) (i) Find the DFA equivalent to the NFA for which the state table is

given in the following and 2

s is the accepting state. (7)

I

S

f

a b

0s

10, ss

2s

1s

0s

1s

2s

1s

10, ss

(ii) Write a note an finite state automata. (6)

PART C — (1 15 = 15 marks)

16. (a) (i) Obtain PDNF of RQPP ~~ (8)

(ii) Obtain PCNF of .~ PQRP (7)

Or

(b) Write each of the following in symbolic from by assuming that the

universe consists of literally everything. (15)

(i) All men are giants.

(ii) No men are giants

(iii) Some men are giants

(iv) Some men are not giants.

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