discrete mathematical structures jan 2010

7/28/2019 Discrete Mathematical Structures Jan 2010

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USN I I I I I I I I I I I 06CS34Third Semester B.E. Degree Examination, Dec.09/Jan.10

Discrete Mathematical StructuresTime: 3 hrs. Max. Marks:lOO

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Note : A nswer any FI VE fu ll questitms, selectingat least TWO questions rom eaclt p art.

PART- Aa. Let x be the set of all three digit integers that is x = {x is an integer I I 00 5. x 5. 999}. IfAi is

the set of numbers in x whose ith digit is i, compute the cardinality of the set A.vAzu A3-b. Using the laws ofse t theory, simplify each of the following:

i) An(B - A) ii) (AuB)nCvBc. State and prove Demorgan's laws set theory.d. Among the integers l - 200 fmd the numbers of integer 's that are:i) Divisible by 2 or 5 or 9 ii) Not divisible by 5iii) Not divisible by 2 or 5 or 9 iv) Divisible by 5 or not by 2 or 9a. Define tautology S.T

(05 Marks)

(05 Marks) .(OS Marks)

(OS Marks)

((pvq) n ( p -H) " ( q __.. r)] - H is a tautology by constructing truth table. (OS Marks)b. Prove the following by using laws oflogic :

i) p __.. (q __.. r) (p "q) __.. r ii) [- p " ( - qM) v [(qM) v (p M)] r (OS Marks)c. Verify the principles of duality for the logical equivalence :

i) - (p "q) __.._ pv(- p v q) ~ - p v q ii) P.T.(p" (p __.. q) 1\T] => (p v q) __.. r (05 Marks)d. Check whether the following is a val id argument:

Ifl study, then I will not fail in the examination. If I do no t watch T.V. in the evening I willstudy.I failedinthe examination (OS Marks):. I must have watch T.V. in the evening.

a. Define an open statement. Write down the negation of following statements:i) For all integer 'n ' ifn is not divisible by 2 then n is odd.ii) Ifk, m, n are any integers when (k - m) and (m - n) are odd then (k - n) is even.(OS Marks)b. For the universe of all integers, defme the following open statements:Let p(x) : x > 0, s(x) : xis divisible by 3, q(x) : xis even, r(x) : x is a perfect square, t(x) : xis divisible by 7. Write down the following quantified statements in symbolise form:i) At least one integer is even ii) Some even integers are divisible by 3iii) For every integer is either even or odd

iv) Ifx is even and a perfect square then xis not divisible by 3v) Ifx is odd or is not divisible by 7 then xis divisible by 3. _ (OS Marks)c. i) PT 3 xq(x) follows logically from the premises \ixp(x) __.. q(x) and 3xp(x)

ii ) PT the following argument is valid where 'C' is the specification element of universeiix(p(x) __.. q(x)]iix[q(x) __.. r(x)] (05 Marks)-r(c)/ : .-p(c)

d. Give direct proof of the statement if n is odd integer and n2 is an odd integer. (OS Marks)

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7/28/2019 Discrete Mathematical Structures Jan 2010

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06CS344 a. Prove by mathematical induction that for all (+)ve integer n ~ 1,

_ n(n+l}1+ 2 +3 + . .... + n - (OS Mark.t)2b. A sequence (a.} is defined successively by a1 = 4, a.. = a, .1+n, for n ~ 2 find an explicitform. (OS Mark.t)c. The fibonacci numbers are defined recursively by Fo = 0, F1- I and F.= Fn-t+Fn-2 for n 2.Evaluate F2 to F10. (OS Mark.t)d. The Ackermann's number. Am.n arc defined recursively form, n eN as follows:

Ao,n =n+ I, for n 0Am.o = Am-1.1 form :2: 0Am,n = Am-l.p where p = Am.nl form m, n ~ 0P.T. A1 .,- n + 2 for all n eN (OS Mark.t)

PART-B5 a. Let A = {I 2 3 4}. Let 'R ' be a relation on A defined by xRy iff x/y andy= 2x write down:

i) 'R ' is a relation of set ofordered pairsii) Draw di-graph ofRiii) Determine in degrees and out-degrees of a diagraph (OS Mark.t)b. Define the following terms and give an example for each:i) Reflexive ii) Irreflexive iii) Anti symmetriciv) Transitive v) Symmetry (OS Mark.t)c. Let R be an equivalence relation on A and (ab)eA then ST the following statement are:

i) a e [a] or a e R(a) i i) aRb iff, [a]= [b] or R(a) =R(b)iii) [a] n [b] '#+then [a]= [b) (OS Mark.t)d. Draw the hasse diagram respresented by the positive divisor of 36. (OS Marks)

6 a. Let A =B {I 2 3 4}i) f={l 1}(2 3)(3 4)(42) ii)JctA a {abc}.B = {12 3 4} f = {(a l){b I) (c4)}iii) A= {I 2 3 4}, b ={abc d}, f= {(I a) (2 a) (3d) (4 c)} .Determine r is one-one or on to. (OS Mark.t)b. Ifm and n are positive integers with I S n 5 m, P.T. s(m +I , n) = s(m, n - I ) ~ n s(m, n)

(OS Mark.t)c. If f : ~ B is invertible then it has unique inverse. {OS Mark.t)d. State Pigeon-hole principle. If (n + I) number are chosen from I to 2n at least one pair addto (2n + I). (OS Mark.t)7 a. Define Abelian group. Prove that a group G is abelian if and only if for all a, b e G,(a, b)" 1 =a"1b'1 (OS Mark.t)b. Define a cyclic group,show that every cyclic group is abelian but the converse is not true.

(OS Mark.t)e. State and prove Lagrange's theorem. (OS Mark.t)d. Let f: G ~ H be a homomorphism from G into H. If G is abelian, prove that H is alsoabelian. (OS Marks)8 a. Prove that every field is an integral domain.

b. S is a subring ofR ifand only if :i) For all a, beS , we have a+ b eS and abeS ii) For all aeS, we have -aeSc. Show that Zs is an integral domain.d. Prove that Zn is a field ifand only ifn is a prime.

(05 Marks)(OS Mark.t)(05 Mark.t)(OS Mark.t)


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discrete mathematical structures jan 2010

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