discrete mathematics

Discrete Mathematics

EXERCISES

Lecture 11

Dr.-Ing. Erwin Sitompulhttp://zitompul.wordpress.com

11/2Erwin Sitompul Discrete Mathematics

Homework 10, No.1Take a look at the graphs (a), (b), and (c).Determine whether each graph is an Eulerian graph, semi-

Eulerian graph, Hamiltonian graph, or semi-Hamiltonian graph.

Give enough explanation to your answer.


Solution of Homework 10 There are two vertices of odd degree, the

others are of even degree Euler path. Euler path must starts from one vertex of odd

degree and finish at the other vertex of odd degree.

It can also be proven that graph (a) contains a Hamilton circuit.

All vertices have even degree Euler circuit.

Graph (b) contains Hamilton path, i.e., from down left vertex to top left vertex, or the opposite way around.


Solution of Homework 10 There are more than two vertices of odd

degree graph (c) does not contain either Euler path or Euler circuit.

Graph (c) contains Hamilton circuit.


Homework 10, No.2

A department has six task forces. Every task force conducts a routine monthly meeting. The member of the six task forces are: TF1 = {Amir, Budi, Yanti} TF2 = {Budi, Hasan, Tommy} TF3 = {Amir, Tommy, Yanti} TF4 = {Hasan, Tommy, Yanti}TF5 = {Amir, Budi} TF6 = {Budi, Tommy, Yanti} (a) What is the minimum number of time slots that must be allocated so that everyone that belong to more than one task force can attend the meetings that he/she must join without any time conflict?(b) Draw the graph that represents this problem and explain what do a vertex and an edge represent.


Schedule 1

Schedule 2

Schedule 3

Schedule 4

Schedule 5 Schedule 4

Solution of Homework 10

TF1 = { Amir, Budi, Yanti } TF2 = { Budi, Hasan, Tommy } TF3 = { Amir, Tommy, Yanti } TF4 = { Hasan, Tommy, Yanti }TF5 = { Amir, Budi } TF6 = { Budi, Tommy, Yanti }

Amir Budi Yanti Hasan Tommy

TF1 1 1 1 0 0

TF2 0 1 0 1 1

TF3 1 0 1 0 1

TF4 0 0 1 1 1

TF5 1 1 0 0 0

TF6 0 1 1 0 1

TF1

TF6

TF5

TF4

TF3

TF2

vertex task forceedge someone is a member of two task forces at the same time


Exercise 1Substitute the following switch circuit with a simpler equivalent circuit.

Solution:The switch circuit above can be expressed by the following

logical notation: (AB’) (AB) C

(AB’) (AB) C = ((AB’) (AB)) C Associative Law = (A(B’B)) C Distributive Law

= (AT) C Negation Law= A C Identity Law


Three best friends, Amir, Budi, and Cora are talking about the grades that Dudi got in the last semester.

Amir says, “Dudi got at least four A’s.” Budi says, ”No, Dudi got less than four A’s.” “I think,” Cora says, “Dudi got at least 1 A”

If only one of the three best friends said the truth, how many A’s did Dudi get?

Exercise 2

Solution:If Amir said the truth, then Cora should also say the truth.If Cora said the truth, then Amir or Budi should say the truth also.Thus, only Budi said the truth while Amir and Cora did not say the truth (“Dudi got less than four A’s.”).The answer: Dudi did not got any A.


Prove that (X – Y) – Z = X – (Y Z)

Exercise 3

Solution:(X – Y) – Z = (X – Y) Z’ Definition of Difference

= X Y’ Z’ Definition of Difference= X (Y’ Z’) Associative Law= X (Y Z)’ De Morgan’s Law= X – (Y Z) Definition of Difference


Exercise 4

By using the Inclusion-Exclusion Principle, determine the number of positive integers ≤ 300 divisible by 2 or 3.

Solution:SupposeU = Set of positive integers ≤ 300,A = Set of positive integers ≤ 300 divisible by 2,B = Set of positive integers ≤ 300 divisible by 3.ThenA B = Set of positive integers ≤ 300 divisible by 2 and 3,A B = Set of positive integers ≤ 300 divisible by 2 or 3.

A = 300/2 = 150B = 300/3 = 100 A B = 300/6 = 50 divisible by 2 and 3 ≡ divisible by 6

A B = A + B – A B = 150 + 100 – 50

= 200


Exercise 5

Determine whether the following relations are reflexive, transitive, symmetric, or anti-symetric:a) “The sister of”b) “The father of”c) “Having the same parents as”

Solution:a) “The sister of”

Not reflexive One cannot be the sister of him/herself.Not transitive If X is the sister of Y, and Y is the sister of Z, it does not mean that X is the sister of Z (counting step sister as a sister).Not symmetric X is the sister of Y, Y does not have to be the sister of X, since Y can be the brother of X.Not anti-symmetric It can occur that X is the sister of Y, and Y is the sister of X, while X ≠ Y.


Exercise 5

Solution:b) “The father of”

Not reflexive One cannot be the father of him/herself.Not transitive If X is the father of Y, and Y is the father of Z, then X is the grandfather of Z.Not symmetric If X is the father of Y, then it is impossible for Y to be the father of X.Anti-symmetric No violation against the rule.



Exercise 5

Solution:c) “Having the same parents as”

Reflexive Although sounds strange, it is true.Transitive If X R Y, and Y R Z, then X R Z. Symmetric X R Y, then Y R X.Not anti-symmetric X R Y, and Y R X, while X ≠ Y



Exercise 6

Prove that 89 and 55 are relatively prime.

Solution: 89 = 155 + 34 (1) 55 = 134 + 21 (2) 34 = 121 + 13 (3) 21 = 113 + 8 (4) 13 = 18 + 5 (5) 8 = 15 + 3 (6) 5 = 13 + 2 (7)

3 = 12 + 1 (8)

89 and 55 are relatively prime because their GCD(89,55) =1.


Exercise 7

Determine one pair of integers (u,v) which is the solution for 89u + 55v = 8.

Solution: 89 = 155 + 34 (1) 55 = 134 + 21 (2) 34 = 121 + 13 (3) 21 = 113 + 8 (4)

(4) 8 = 21 – 113 (5)(3) 13 = 34 – 121 (6)

(6)(5) 8 = 21 – 1(34 – 121) 8 = 221 – 34 (7)

(2) 21 = 55 – 134 (8)(8)(7) 8 = 221 – 34

8 = 2(55 – 134) – 34 8 = 255 – 334 (9)


Exercise 7

Solution: 89 = 155 + 34 (1) 55 = 134 + 21 (2) 34 = 121 + 13 (3) 21 = 113 + 8 (4)

8 = 255 – 334 (9)

(1) 34 = 89 – 155 (10)(10)(9) 8 = 255 – 334

8 = 255 – 3(89 – 155) 8 = 555 – 389

Thus, one possible pair of integers (u,v) as the solution of the linear combination is (–3,5).

Determine one pair of integers (u,v) which is the solution for 89u + 55v = 8.


Exercise 8

The ISBN code of a printed book is ISBN-13: 978-051A0B2934.

If B mod A = 2, determine A and B.Solution:Processing the first 12 numbers of the code: 91 + 73 + 81 + 03 + 51 + 13 +

A1 + 03 + B1 + 23 + 91 + 33 = 70 + A + B.

Including the check digit (the 13th number):70 + A + B + 4 0 (mod 10)74 + A + B 0 (mod 10)A + B = {6,16,26,36,…}

WhileB mod A = 2Possible combinations for (A,B) are (3,5), (4,6), (5,7), (6,8), (7,9), and (3,8).

The combination that fulfills the condition is: A = 7 and B = 9, where A + B = 16.


Exercise 9

The fixed-line phone numbers in one region consist of 8 digits. The first digit may not be 0 or 1.(a) How many possible phone numbers are there in the region? (b) How many phone numbers have no 0?(c) How many phone numbers have at least one 0?

Solution:(a) 810101010101010 = 80.000.000 phone numbers.

(b) 89999999 = 38.263.752 phone numbers.

(c) Phone numbers with at least one 0 = Possible phone numbers –

Phone numbers without any 0= 41.736.248 phone numbers.


Exercise 10

(a) Determine the number of ways a president can fill the position of Foreign Minister, Minister for Internal Affairs, Defense Minister, and Secretary of State, out of 45 candidates that he has?.

(b) In how many ways can you choose 4 pails of wall paint out of 45 pails of wall paint, each with different colors?

Solution:(a)

(b)

45!(45,4)

(45 4)!P

3.575.880 ways

45!(45,4)

(45 4)!4!C

148.995 ways


7! 9!

(7 2)!2! (9 3)!3!

Exercise 11

Persib Bandung conducts a preparation training center with 16 players for the next Indonesian Super League season. The players are requested to choose 5 people among them to become the member of team council for occasional negotiations with the management.(a) In how many ways can the players choose their team council?(b) If 7 of the 16 players are young and below 23 years old, in

how many ways can the team council be elected if there are 2 young players in the council?

Solution:(a)

(b)

16!(16,5)

(16 5)!5!C

4.368 ways

(7,2) (9,3)C C 1.764 ways


End of the Lecture

discrete mathematics

Documents

truth dudi

graph b

degree euler circuit

ab ab c ab ab c

x y zassociative law

vertices of odd degree

x y z exercise

vertex of odd degree