Discrete Mathematics

Discrete Mathematics EXERCISES Lecture 11 Dr.-Ing. Erwin Sitompul http://zitompul.wordpress.com

Homework 10, No.1 Take 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.

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} vertex task force edge  someone is a member of two task forces at the same time

Exercise 1 Substitute the following switch circuit with a simpler equivalent circuit. Solution: The switch circuit above can be expressed by the following logical notation: (AB’)(AB)  C (AB’)(AB)  C = ((AB’)  (AB))  C Associative Law = (A(B’B))  C Distributive Law = (AT)  C Negation Law = A CIdentity Law

Exercise 2 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? 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.

Exercise 3 Prove that (X – Y) – Z = X – (Y Z) 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: Suppose U = Set of positive integers ≤ 300, A = Set of positive integers≤ 300 divisible by 2, B = Set of positive integers ≤ 300 divisible by 3. Then A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 = 150 B = 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 Ycan 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 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: 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 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: c) “Having the same parents as” Reflexive Although sounds strange, it is true. Transitive  If XRY, and YRZ, then XRZ. Symmetric  XRY, then YRX. Not anti-symmetric  XRY, and YRX,while X ≠ Y

Exercise 6 Prove that 89 and 55 are relatively prime. Solution: 89 = 155 + 34 (1) 55 = 134 + 21 (2) 34 = 121 + 13 (3) 21 = 113 + 8 (4) 13 = 18 + 5 (5) 8 = 15 + 3 (6) 5 = 13 + 2 (7) 3 = 12 + 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 = 155 + 34 (1) 55 = 134 + 21 (2) 34 = 121 + 13 (3) 21 = 113 + 8 (4) (4) 8 = 21 – 113 (5) (3) 13 = 34 – 121 (6) (6)(5) 8 = 21 – 1(34 – 121) 8 = 221 – 34 (7) (2) 21 = 55 – 134 (8) (8)(7) 8 = 221 – 34 8 = 2(55 – 134) – 34 8 = 255 – 334 (9)

Exercise 7 Determine one pair of integers (u,v) which is the solution for 89u + 55v = 8. Solution: 89 = 155 + 34 (1) 55 = 134 + 21 (2) 34 = 121 + 13 (3) 21 = 113 + 8 (4) 8 = 255 – 334 (9) (1) 34 = 89 – 155 (10) (10)(9) 8 = 255 – 334 8 = 255 – 3(89 – 155) 8 = 555 – 389 Thus, one possible pair of integers (u,v) as the solution of the linear combination is (–3,5).

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: 91 + 73 + 81 + 03 + 51 + 13 + A1 + 03 + B1 + 23 + 91 + 33 = 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,…} While B mod A = 2 Possible 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) 810101010101010 = 80.000.000 phone numbers. (b) 89999999 = 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)

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)

End of the Lecture

Discrete Mathematics