Math161 HW#2, DUE DATE:

Transcription

1 Math161 HW#2, DUE DATE: Question1) How many students in a class must there be to ensure that 6 students get the same grade (one of A, B, C, D, or F)? Question2) 6 computers on a network are connected to at least 1 other computer Show there are at least two computers that are have the same number of connections Question3) In a group of 8 people show that at least two have their birthday on the same day of the week. Question4) A box contains 10 French books, 20 Spanish books, 8 German books, 15 Russian books, and 25 Italian books. How many must we choose to ensure that we have 12 books in the same language? Question5) There are 30 students in a class. While doing a keyboarding test one student made 12 mistakes, while the rest made fewer mistakes. Show that at least 3 students made the same number of mistakes. Question5) A canteen has 95 tables with a total of 465 seats. Can we be sure that there is a table with at least 6 seats? Question6) In a group of 100 people, several will have their birthdays in the same month. At least how many must have birthdays in the same month? Question7) In a class of students undergoing a computer course the following were observed. Out of a total of 50 students: 30 know Pascal, 18 know Fortran, 26 know COBOL, 9 know both Pascal and Fortran, 16 know both Pascal and COBOL, 8 know both Fortran and COBOL, 47 know at least one of the three languages. From this we have to determine a. How many students know none of these languages? b. How many students know all three languages? Question8) In a group of 15 pizza experts, ten like Canadian bacon, seven like anchovies, and six like both. a) How many people like at least one of these toppings? b) How many like Canadian bacon but not anchovies? c) How many like exactly one of the two toppings? d) How many like neither? Question9) Among the 30 students registered for a course in discrete mathematics, 15 people know the JAVA programming language, 12 know the HTML, and 5 know both of these languages. a) How many students know at least one of JAVA or HTML? b) How many students know only JAVA? c) How many students know only HTML?

2 d) How many students know exactly one of the languages JAVA or HTML? e) How many students know neither JAVA nor HTML? Question10) How many integers from 1 to 1000 are either multiples of 3 or multiples of 5 Question11) In a certain state, license plates consist of from zero to three letters followed by from zero to four digits, with the provision, however, that a blank plate is not allowed. a. How many different licence plates can the state produce? b. Suppose 85 letter combinations are not allowed because of their potential for giving offense. How many different license plates can the state produce? Question12) How many positive integers less than or equal to 100 are relatively prime to 15? Question13) In a recent survey of college graduates, it was found that 200 had undergraduate degree in arts, 95 had undergraduate degrees in science, and 120 had graduate degrees. Fifty-five of those with undergraduate arts degrees had also a graduate degree, 40 of those with science degrees had graduate degrees, 25 people had undergraduate degrees in both arts and science, 5 people had undergraduate degrees in arts and science and also graduate degrees. a) How many people had at least one of types of degree mentioned? b) How many people had an undergraduate degree in science but no other degree? Question14) Suppose that F = Phoebe likes french fries. C = Phoebe likes chili. P = Phoebe likes pizza. (a) Translate the following logical statements into words: (i) P C. (ii) (F C) P. (iii) P (C F). (b) Translate the following statements into logical symbols: (i) Phoebe likes pizza if Phoebe likes French fries. (ii) Phoebe likes chilli and either Phoebe likes French fries or Phoebe likes pizza. (iii) If Phoebe likes French fries or Phoebe doesn t like pizza, then Phoebe likes chilli. Question15) Construct a truth table for the following compound statements: a) (P Q) ( P Q) b) P (P Q). c) ( p q) ( p q)

3 d) ( p q) ( p q) e) ( p q) q f) p ( p q) g) ( p q) ( ( p) r) ( q r) h) (P Q) (Q R). Question16) Determine whether the following compound statement is a tautology, contradiction or contingency: p p q a) ( ( )) p b) (P Q) (Q P) c) (( p q) ( p r) ) ( q r) ( ) d) ( p q) ( p ( q) ) e) ( p q) (q ( p q)) Question17) Show that the following compound statements are logically equivalent: a) P Q and P Q b) (( p q) ( p) ) and (( p q) ) ( p q p q ) and (( p ( q) ) (( p q) )) c) ( ) ( ( )) Question18) Use Karnaugh maps to simplify the following expressions: Note: p = p = p are alternative notations for the negation of p.

4 e) α ( x, y, z) = xyz + xyz + xy z + x yz + x yz f) α ( x, y, z, w) = xyzw + xyzw + xyzw + xyzw + xyzw + xyzw g) α ( x, y, z) = x y z + x y z + xy z + xyz + xyz Question19) Construct the circuit diagrams corresponding to the following Boolean expressions: Question19) Consider the following circuit. a. Complete the truth table. Question20) a) Give a sum-of-products Boolean expression corresponding to the truth table at right. x y z out b. Simplify your expression. c) draw a circuit corresponding to your expression. Question21) x y z Output

5 a) Use the truth table; find the Sum-of-Products Boolean expression. b) Construct the Circuit c) Draw the K-map. d) Find the minterm of the Sum-of-Products Boolean expression. Question22) Complete the following for SUM. Draw neatly to get credit. Draw dots on the grid where connections should be. For the truth tables below, express the minterm sum of products, and the maxterm product of sums: A B C output Question23)

6 a) Write the Boolean Expressions that represent the following circuit. b) Find the minimized sum-of-product Boolean expression. AB AB A B A B C C 1 1 Question24) Given the following graph, a) Find the number of vertices and edges. b) Find the degree of all vertices. c) Find Euler cycle if it exists. d) Find Euler path if it exists. Question 25) i) use the graph ii) use the adjacency matrix

7 Question 26) Find an Euler cycle on this graph if possible Question 27) Find an Euler path on this graph if possible Question 28) Find an Euler path on this graph if possible Question29) Given the graph

8 Question30) Find an Euler cycle for the given graph, if possible. If not possible, explain why. If it is possible number the edges of your path so I can easily follow it. Question31) Answer the questions about an (undirected) graph G = (V A), where V = A = (1 3) (1 5) (1 7) (1 9) (1 10) (2 3) (2 4) (2 6) (2 8) (3 4) (3 8) (3 10) (4 5) (4 8) (4 9) ( 5 6) (5 8) (6 9) (7 8) (8 10) (a) Show the graph is connected. (b) Give the degrees of each of the edges. (c) Find a cycle that includes nodes 1 and 4. (d) Is there an Euler cycle through the graph? Why or why not? (e) Is there an Euler path through the graph? Why or why not? Question32) Consider a graph with vertex set V={1,2,3,4,5,6}, and arc set A={(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(2,3),(3,4),(4,5),(5,6),(6,1)} Does there exist an Eulerian cycle in this graph? Why or whynot? Question33) Given A = a. Draw the graph for which A is the adjacency matrix. b. Find the number of paths of length 4 from 5 to 3. c. Find the total number of cycles of length 6 in this graph.