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Exam in Discrete Mathematics

First Year at The TEK-NAT Faculty

June 11th, 2014, 9.00-13.00

ANSWERS

Part I ("regular exercises")

Exercise 1 (6%).

Find the expansion of(2xy)4using The Binomial Theorem.

Answer:16x432x3y+24x2y28xy3+y4

Exercise 2 (8%).

Find witnesses proving thatf(x) =2x3+x2+5 isO(x3).

Exercise 3 (12%).

1. Use the Eu clideanalgorithm to find the gr eatestcommon divisor of 46 and 21.

Answer:1

2. Find integ erssandtsatisfying that gcd(46,21) =s46+t21.

Answer: s=5,t=11

3.

Determine all integers xsuch that

x2(mod 46)andx1(mod 21).

Answer: x232(mod 966)

Page 1 of 7

Exercise 4 (9%).

Prove by induction that

nå i=1(4i+1) =2n2+3n, for every positive integern.

Exercise 5 (6%).

1. Constr ucta t ruthtable for the compound pr oposition(p^ :q)!(r_q).

Answer:p q r(p^ :q)!(r_q)T T TT

T T FT

T F TT

T F FF

F T TT

F T FT

F F TT

F F FT

2. Is the compo undpr opositionin question 1 a tautology?

Answer:No.a

b c d e f g h i jkm nFigure 1: A graphGconsidered in Exercise 6.

Page 2 of 7

Exercise 6 (10%).

A graphGwith 13 edges is shown in Figure 1. The edges ofGhave weights given by the following tableEdgeabcdefghijkmn

Weight1133645624272

1. Use Prim"s algorithm to find a minimum spanning tr eeSinG. Write the edges ofSin the order in which they are added toSby Prim"s algorithm. (If there is more than one possible solution then write only one of them.)

One possible solution:a, b, i, n, k, c, e

2. Use Kr uskal"salgorithm to find a minimum spanni ngtr eeTinG. Write the edges ofTin the order in which they are added toTby Kruskal"s algorithm. (If there is more than one possible solution then write only one of them.)

One possible solution:a, b, i, k, n, c, e

Exercise 7 (9%).

LetA=fa,b,c,dgand letR=f(a,b),(b,c),(c,d),(d,b)gbe a relation onA. 1.

Draw the dir ectedgraph r epresentingR.

Answer:b

cda2.Determine the transitive closur eRofR.

Answer:

R

Page 3 of 7

3.Determine a matrix MRrepresentingR.

Answer:

2 6

640 1 1 1

0 1 1 1

0 1 1 1

0 1 1 13

7 75

Exercise 8 (10%).

A setSis defined recursively by

Basis step:02S

Recursive step:ifa2Sthena+32Sanda+52S.

1.

Determine the set S\ fa2Zj0

Answer:f3,5,6,8,9,10,11g

2.

Pr ovethat e veryinteger a8 is contained inS.

Page 4 of 7

Part II ("multiple choice" exercises)

Exercise 9 (10%).

Letf(x) = (x2+5x+3)(x+2logx),forx>0.Answerthefollowing5true/false exercises

1.f(x)isO(x4).

TrueFalse

2.f(x)isO(x3).

TrueFalse

3.f(x)isO(x2).

TrueFalse

4.f(x)isO(x3logx).

TrueFalse

5.f(x)isO(x2logx).

TrueFalse

Page 5 of 7

Exercise 10 (6%).

LetA=f1,3,5gandB=f3,4,5gbe sets.

1.

What is the car dinalityof the power set P(A[B)

48163264

2.

Which of the following ar eelements of AB?

f1,3g(1,3)(4,5)(5,5)

Exercise 11 (8%).

Consider the following algorithm:

Proceduresum(n: positive integer)

s:=0 fori:=1ton .forj:=1toi . s:=s+j returns 1. Suppose that pr oceduresum is started with input n=4. Then what num- ber is returned by the algorithm?

10204045

2. The worst -casetime complexity of pr oceduresum is:

O(n)O(nlogn)O(n3/2)O(n2)

Page 6 of 7

Exercise 12 (6%).

Let M R=2 6

66641 0 0 1 1

0 1 0 0 0

1 0 1 1 1

1 0 1 1 0

0 0 1 0 13

7 7775
exercises

1.Ris reflexive.

TrueFalse

2.Ris symmetric.

TrueFalse

3.Ris antisymmetric.

TrueFalse

Page 7 of 7

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