Acharya Nagarjuna University (ANU) 2006 M.C.A III - Discrete Mathematics. - Question Paper

M.C.A. Degree Examination, December 2006
First Year
Paper III - Discrete Mathematics.

[DMCA 108)

M.C.A. DEGREE EXAMINATION, DECEMBER 2006 First Year

Paper-VIII: DISCRETE MATHEMATICS Time : Three hours    Maximum : 75 marks

SECTION A - (3 X 15 = 45 marks)

Answer any THREE of the following.

1.    (a} Co nsi de r t h e followi ng sets:

= {l} 5 = {U) C = {1,5,9} J? = {l,2,3,4,5} ={1,15,7,9} t/ = {l,2,....S,9} Insert the correct symbol :z or x between each pair of sets.

(b) Construct the truth table for {P

2.    (a) Determine types of relations with examples.

(b) Prove the set I of all real numbers between 0 and 1 is uncountable.

3.    (a) Show that the set N of natural numbers is a semigroup under the operation = maxfi, j}. Is it a monoid?

(b) Prove that for any commutative monoid (w,*), the set of idem potent elements of M forms a sub monoid.

4.    Prove that every row or column in the composition table of a group (G\*) is a permutation of the elements of G.

5.    (a) If (?,*)is an abelian group_; then for all a,b e G show that (a *by =<f *b^B.

(b)    Show that if every element in a group is its own inverse, than the group must be abelian.

SECTION B - (5 X 5 = 25 marks)

Answer any FIVE of the following.

6.    Explain Hasse Diagram.

7.    Show that the proportions    and -ipv-iq are logically equivalent.

8.    Consider the relation R = {(l,l)(l_s2)(l_r3)(3,3)} on the set A = {l,2,3}. Determine whether or not the relation R is

(a) R efI exive    (b) S ym m et ri c

(c)    Transitive    (d) Anti symmetric

9.    Find the domain D of each of the following real-valued functions.

{3)    =

x-2

<b> f(x) = x- -3x-4

W 27

10.    Let r'={l_r2,3,4} and let / = {(13)(2,1)(3_S4)(4,3)} and _s = {(UX2,3)(3,1)(4.1)| and

{a) fog

(b)    gof

(c)    fof

11.    Let A = (1.2,3,4,5). Determine the truth value of each of the following statements.

{a) @xeA){x + 3 = lQ)

<b> (weJ)(j + 3 <10)

{c} (3xX-T + 3<5)

(d)    (vxe/l)(;r-i-3<7)

12.    Let S be a semigroup with identity e. and let b and b¹ be inverses of a. Such that l>=b

13.    Explain Lagrange's Theorem.

SECTION C - (S X 1 = 5 marks)

Answer ALL of the following:

14.    What is a function?

15.. What is structure?

16.    Consider the ring Z_s = {0,1,2,......9} of integers modulo 10. Then find the units of Z_]S.

17.    Rewrite the following statement using the conditional. If it is cold, he wears a hat.

18.    Define semigroup homomorphism.

Attachment: acharya-nagarjuna-university--anu--2006-m-c-a--8401-1.pdf

Earning:   Approval pending.