1. Let pand be the propositions I bought a new book this week and "I read non- fiction books every weekend" respectively. Express each of these compound propositions as an English sentence. b. pV4 6.14 d. P e. f. P. 2. Determine the truth-values of the following: If P(x): "The sentence x contains at most 2 times the word (the)". 1. PLook at the moon!) 2. P/She came back!) b. If Q(x,y): " 5x + y is divisible by 5 ** 1.064.5) 2.0(3,4) 3. Translate the given statement into propositional logic using the propositions provided. 4. You can see the movie only if you are over 18 years old or you have the permission of a parent. Express your answer in terms of me: "You can see the movie," e "You are over 18 years old," and p: "You have the permission of a parent". 6. "To use the wireless network in the airport you must pay the daily fee unless you are a subscriber to the service "Express your answer in terms of w:"you can use the wireless network in the airport". d: "you pay the daily fee". s: "You are a subscriber to the service 4. Determine whether the following conditional and biconditional are true or false. 4. If 1 + 1 - 3, then unicorns exist . 5 (PA) FF FT TF TIT h (p V)-(-A) P pv PA (p Va) -- PA) FFF F FT FT F F T T T F F T F T TT F TTT - 7. Find the negation of each of the following statements. a James is young and strong: b. Rita will move to Oregon or Washington, cif' it rains today then the home team wins: d xayP(x, y) A VxVyQ(x, y). e Exay(O(x, y) + Oly, x)). 1. Wy9x32(T (x, y, z) VQ(x, y). 8. Are these system specifications consistent? "Whenever the system software is being upgraded users cannot access the file system". "If users can access the file system, then they can save new files". "If users cannot save new files, then the system software is not being upgraded". 9. State the converse, contrapositive, and inverse of each of these conditional statements. a. When I stay up late, it is necessary that I sleep until noon. b. I climb mountains whenever it is a sunny summer day. If you drink Pepsi, then you are happy. 10. Select the statement that is logically equivalent to: "If you don't carry an umbrella, you'll get soaked." a. You carry an umbrella and you won't get soaked. b. You carry an umbrella or you get soaked. c. You don't carry an umbrella and you get soaked. d. You don't carry an umbrella or you get soaked. Page 3 of 5 11. Show that each of these conditional statements is a tautology by without using truth tables 4. (p14) - b. p (p V d. [-)--))--(p) 12. Show that pq) and (p> A) :Wy6y=0). 3x3)(x + y + x). d) Vxx#0 - 3y = 1)). Page 4 of 5 17. Express each of these system specifications using predicates, quantifiers, and logical connectives. a) Every user has access to an electronic mailbox b) The system mailbox can be accessed by everyone in the group if the file system is locked c) The firewall is in a diagnostic state only if the proxy server is in a diagnostic state 18. Are these system specifications consistent? "Whenever the system software is being upgraded, users cannot access the file system. If users can access the file system, then they can save new files. If users cannot save new files, then the system software is not being upgraded." 19. Question 12 Let I (x) be the statement "x has an Internet connection" and C(x, y) be the statement "x and y have chatted over the Intemet," where the domain for the variables x and y consists of all students in your class. Use quantifiers to express each of these statements. a. Jerry does not have an internet connection, b. Rachel has not chatted over the Internet with Chelsea. c. Jan and Sharon have never chatted over the Internet. d. No one in the class has chatted with Bob e Everyone in your class with an Internet connection has chatted over the Internet with at least one other student in your class. 20. Determine whether each of these compound propositions is satisfiable. a. (V) (OV) (PV-) b. (Q) () 21. Find a counter example, if possible, to these universally quantified statements, where the domain for all variables consists of all integers. a. VxVy(x-y-+*) b. Vxay(y -x). c. VxVy(xy 2x) 22.. Find the bitwise OR, bitwise AND, and bitwise XOR of each of these pairs of bit strings. a) 101 1110, 010 0001 b) 1111 0000, 1010 1010 Page 5 of 5