Quiz 1 ID#: Name: 1. p, q, r (Let p, q and r be propositions. Determine whether the following equation holds or not by completing the truth table belo

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1 Quiz 1 ID#: Name: 1. p, q, r (Let p, q and r be propositions. Determine whether the following equation holds or not by completing the truth table below.) (p q) r p ( q r). p q r (p q) r p ( q r) x T T T T T T F T T F T T T F F T F T T T F T F F F F T F F F F T [ (Conclusion)] 2. p ( q r) p (q r) (p q) r (Fill some of the underlined blanks below by to express (p q) r.) (p q) r p ( q r) 3. x,, (Fill each underlined blank with, or to express x in the truth table above. There may be voids.) x ( p ( q r)) ( p ( q r)) Message 25 What is your dream? Describe your vision of yourself and the world 25 years from now. HP If you wish, write Do Not Post.

2 Solutions to Quiz 1 1. p, q, r (Let p, q and r be propositions. Determine whether the following equation holds or not by completing the truth table below.) (p q) r p ( q r). p q r (p q) r p ( q r) x T T T F T T T T F T F T T F T T T T T T F F T T T T T F F T T F T F F T T F T F T T F T F T F T T T F T T T T F F F T T F T T F F T T T F T F T F T T F F T T T F T T F T F T T T T F T F F F T T T T F T F F F T F F F F F T T F F F F F T T F T T F T T F F F F T F F F T T F T F T T F T F T [ (Conclusion)] F (p q) r p ( q r). 2. p ( q r) p (q r) (p q) r (Fill some of the underlined blanks below by to express (p q) r.) p (q r) (p, q, r) (F, T, F ) F T (p q) r (p, q, r) (F, F, T ) F T (p q) r p ( q r). 3. x,, (Fill each underlined blank with, or to express x in the truth table above. There may be voids.) x = ( (p q) r) ( p ( q r)). x ( (p q) r) ( p ( q r)) ( p ( q r)) ( p (q r)). p q r (p, q, r) (F, F, F ) F T y z y z (p q) r p q r, p ( q r) p q r. (p, q, r) (F, F, T ) F (p, q, r) (F, T, F ) F T

3 Quiz 2 ID#: Name: (A) (Explain the impossibility of covering up the room below with 24 mat size usable space (a 25 mat size room with two large pillars at ) by one mat size futon of shape (A) without overlapping.) (A) 2. (a) 20 A, B, C, D, E, F 6 (How many ways are there to sell 20 tickets by 6 assuming that each one sells at least one?) [ Solution only! n C r, n P r, n! (No need to evaluate n C r, n P r or n!.) ] (b) 14 A, B, C, D, E, F 6 (The number of ways to sell 14 tickets by 6, allowing that some may sell no tickets, is equal to the number in the previous problem. Explain why.) Message What is most precious to you? HP If you wish, write Do Not Post.

4 Solutions to Quiz (A) (Explain the impossibility of covering up the room below with 24 mat size usable space (a 25 mat size room with two large pillars at ) by one mat size futon of shape (A) without overlapping.) (A) 25, (a) 20 A, B, C, D, E, F 6 (How many ways are there to sell 20 tickets by 6 assuming that each one sells at least one?) [ Solution only! n C r, n P r, n! (No need to evaluate n C r, n P r or n!.) ] 19C 5 = (b) 14 A, B, C, D, E, F 6 (The number of ways to sell 14 tickets by 6, allowing that some may sell no tickets, is equal to the number in the previous problem. Explain why.) [ : ] A, B, C, D, E, F a, b, c, d, e, f (b) a + b + c + d + e + f = 14 a = a + 1, b = b + 1, c = c + 1, d = d + 1, e = e + 1, f = f + 1 a + b + c + d + e + f = 20 (a, b, c, d, e, f) (a, b, c, d, e, f ) (a, b, c, d, e, f) = (a 1, b 1, c 1, d 1, e 1, f 1) (a + 1, b + 1, c + 1, d + 1, e + 1, f + 1) = (a, b, c, d, e, f )

5 Quiz 3 ID#: Name: 3 (A, B, C ) n 1 1 p(n): 2 n 1 (Hanoi s Tower with n disks can be completed in 2 n 1 steps.) 1. p(6) p(7) (Explain that p(7) is true assuming that p(6) is true.) 2. n = 4 A C A B, C n = 5 (Move from A to C. What is the first move when n = 4 and n = 5? Assume that the total number of steps is minimum.) n = 4 (First move the smallest to) B C ( n = 5 (First move the smallest to) B C ( 3. 1, 2,..., 7 7 A 5,4,3,2,1 B 6 C 7 A (A: 7, B: 5, 4, 3, 2, 1, C: 6. Move A to B or C and complete it with minimum number of moves.) (Move to) B C ( (How many moves?) Message (Anything that made you rejoice, sad or angry, or you are thankful of recently?) If you wish, write Do Not Post.

6 Solutions to Quiz 3 3 (A, B, C ) n 1 1 p(n): 2 n 1 (Hanoi s Tower with n disks can be completed in 2 n 1 steps.) 1. p(6) p(7) (Explain that p(7) is true assuming that p(6) is true.) 7 A B C Step 1. p(6) A = 63 C Step 2. Step 1 A C B A B Step 3. Step 2 A B C 6 p(6) = 63 B C 6 Step 1, 2, 3 (2 6 1) (2 6 1) = = = 127 = A, B, C A, B, C p(7) 2. n = 4 A C A B, C n = 5 (Move from A to C. What is the first move when n = 4 and n = 5?). n = 4 (First move the smallest to) B n = 5 (First move the smallest to) C 1, 2, 3, 4, 5 n = 4 4 C 1 3 B 1 2 C 1 B n = 5 1 C n 3. 1, 2,..., 7 7 A 5,4,3,2,1 B 6 C 7 A (A: 7, B: 5, 4, 3, 2, 1, C: 6. Move A to B or C and complete it with minimum number of moves.) (Move to) B (How many moves?) 95 Step 1. 5 C = 31 B = = 95

7 Quiz 4 ID#: Name: First Initial Last Initial 26 Among 693 April students, there is a pair with the same first and last initial If 20 books were read in 14 days and at least one each day, then the total number of books read between some day to the other is exactly seven. Message

8 Solutions to Quiz First Initial Last Initial 26 Among 693 April students, there is a pair with the same first and last initial. First Initial 26 Last Initial = > If 20 books were read in 14 days and at least one each day, then the total number of books read between some day to the other is exactly seven. 1 a 1 2 a 2 14 a 14 b 1 = a 1, b 2 = a 1 + a 2, b 3 = a 1 + a 2 + a 3, b 14 = a 1 + a a 14 b 1 1 b 2 2 b b 1 < b 2 < < b 14 = 20. (1) 7 8 b < b < < b = 27. (2) 1 b 1, b 2,..., b 14, b 1 + 7, b 2 + 7,..., b b 1, b 2,..., b 14 (1) b 1 +7, b 2 +7,..., b (2) b 1, b 2,..., b 14 b 1 +7, b 2 +7,..., b b i b j +7 b i = b j +7 b i b i = 7 b i, b j i > j b i b i = (a 1 + a a j + a j a i ) (a 1 + a a j ) = a j a i b i b i = 7 7 j + 1 i 7

9 Quiz 5 ID#: Name: 1. S S 10 S Ten couples including Mr. & Mrs. S participated in a party. Mrs. S found out that no one shook hands with one s spouse and everyone (besides Mrs. S) shook hands with different number of people. Explain the following. (a) S Mrs. S shook hands with odd number of people. (b) 18 There is exactly one who shook hands with 18 people. (c) 18 The spouse of the one who shook hands with 18 people shook hands with nobody Knight On 7 7 size board, Knight cannot visit all places once and come back. Message (1) About marriage, family and children. (2) About this course; requests and suggestions for improvement. If you wish, write Do Not Post.

10 Solutions to Quiz 5 1. S S 10 S (Ten couples including Mr. & Mrs. S participated in a party. Mrs. S found out that no one shook hands with one s spouse and everyone (besides Mrs. S) shook hands with different number of people. Explain the following. (a) S Mrs. S shook hands with odd number of people S , 3, 5, 7, 9, 11, 13, 15, 17 9 Theorem S (b) 18 There is exactly one who shook hands with 18 people. S 18 S 18 (c) 18 The spouse of the one who shook hands with 18 people shook hands with nobody. 18 (a) Knight On 7 7 size board, Knight cannot visit all places once and come back Knight

11 Quiz 6 ID#: Name: 1. 8 Consider trees with 8 vertices. (a) 7 2, 2, 2, 1, 1, 1, 1 What is the degree of the remaining vertex if the degrees of seven vertices are 2, 2, 2, 1, 1, 1, 1? (b) 3 Depict three non-isomorphic trees satisfying the condition in the previous problem. 2. A, B, C, D, E, F, G, H 8 2 Find a most inexpensive network connecting A, B, C, D, E, F, G, H and its total cost by referring to the cost table below. Cost Table G F H A B C A B C D E F G H A B C D E F G H E D Total Cost: units Message World Citizen ICU

12 Solutions to Quiz Consider trees with 8 vertices. (a) 7 2, 2, 2, 1, 1, 1, 1 What is the degree of the remaining vertex if the degrees of seven vertices are 2, 2, 2, 1, 1, 1, 1? 8 e 7 2 x x = x = 14 4 (b) 3 Depict three non-isomorphic trees satisfying the condition in the previous problem. 1 4, 2, 2, A, B, C, D, E, F, G, H 8 2 Find a most inexpensive network connecting A, B, C, D, E, F, G, H and its total cost by referring to the cost table below. Cost Table G F H A B C A B C D E F G H A B C D E F G H E D Total Cost: 15 units Connected Total Cost 15 Tree Tree

13 Quiz 7 ID#: Name: 1. Explain that the bottom graph is not Eulerian. 2. a e {a, e}. How many edges does it need to make the graph Eulerian? Describe edges to add. 3. Theorem 7.3 S,, ω( ) Show that the graph is not Hamiltonian by applying Theorem 7.3. Describe S,, and ω( ). a d m b e g f h j i k n c l o Message ICU C

14 Solutions to Quiz 7 1. Explain that the bottom graph is not Eulerian. b 3 Theorem a e {a, e}. How many edges does it need to make the graph Eulerian? Describe edges to add. b.d, f, g, i, j, l, n 4 {b, d}, {f, g}, {i, j}, {l.n} a d m b e g f h j i k n c l o 3. Theorem 7.3 S,, ω( ) Show that the graph is not Hamiltonian by applying Theorem 7.3. Describe S,, and ω( ). S = {c, e, h, k, m} ( ) ω( ) = 6 > 5 = S Theorem 7.3 : a, o S = {a, c, e, h, k, m, o} ω( ) = 8. a d m a d m b e g b e g f h j f h j i k n i k n c l o c l o

15 Quiz 8 ID#: Name: 1. Γ Show that Γ is non-planar. Γ A connected plane graph is 4-regular and every face is either a 3-gon or a 4-gon. Find the number of 3-gons. Message (1) (2) ICU

16 Solutions to Quiz 8 1. Γ Show that Γ is non-planar. Γ v = 11, e = 20 Γ Γ f (Theorem 8.1) 2 = v e + f = f. f = Proposition 8.2 (ii) 40 = 2e = n 1 + n n = 44. Γ K K 5 Γ A connected plane graph is 4-regular and every face is either a 3-gon or a 4-gon. Find the number of 3-gons. v e f 3 t F 1, F 2,..., F f n 1, n 2,..., n f F 1,..., F t 3 F t+1,..., F f (f t ) 4 n 1 = n 2 = = n t = 3 n f+1 = n f+2 = = n f = 4 Proposition 8.2 (i) (ii) 4v = 2e, 2e = n 1 + n n t + n t+1 + n t n f = 3t + 4(f t) = 4f t. Theorem = 2e 4 e + 2e + t = t 4 4. t = (octahedron graph) (cuboctahedron graph)

Page 1 of 6 B (The World of Mathematics) November 20, 2006 Final Exam 2006 Division: ID#: Name: 1. p, q, r (Let p, q, r are propositions. ) (10pts) (a

Page 1 of 6 B (The World of Mathematics) November 0, 006 Final Exam 006 Division: ID#: Name: 1. p, q, r (Let p, q, r are propositions. ) (a) (Decide whether the following holds by completing the truth