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MCA.1/11.12. 1079

M.C.A. DEGREE I SEMESTER EXAMINATION NOVEMBER 2012

CAS 2104 DISCRETE MA THEMA TICAL STRUCTURES

Time: 3 Hours. Maximum Marks: 50

PART A Answer ALL questions)

15 x 2 30)

1. a) Is Ax B~C)=AxB ~ Axe where ~ denotes the symmetric difference? Justifyyour

claim .

b) Write the truth table ofthe disfunction PVQ of the following statements :

3 is an integer

q: 3 is greater than 5

c) Let X CC the set of complex numbers. Define a relation - as follows;a + ib - c + id if a ~ c and b ~ d . Is - a partial order relation? Is it a total order?Justify.

n a) Define a Boolean matrix. How are the usual matrix operations defined on it?

b) Let B = {3nln Z,O 3n ~ 2102}. Find the number of elements in B.

c) State the pigeon hole principle.

l a) Find the recurrence relation and initial condition for the sequenceS: 0,1,3 , 7, 15 2 n -1 .

b) Write the characteristic equation of the following linear homogeneous equation .a n = 7a n_I - 1 0 an_ with initial conditions ao 1, a . = 8 . Also find the roots of thecharacteristic equations.

c) Consider the non homogeneous recurrence relation a n -5 an = 3n n ~ I with initialconditions a o 1. Show that the general solutions In of the equation can be obtained froma particular solution p of the equation and a solution U; of a homogeneous recurren cerelation.

IV. a) Construct the truth table for the Boolean expression a X I X2,X3) X + x2 . x .+ X 2 X ~ ) ).

b) Give an example of a Boolean algebra.

c) In a Boolean algebra B , for any a, band c, prove that a bl ac = a bl

P.T.O)

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