MATH Section 1.2

Let p, q, r be the propositions: p: The user enters a valid password. q: Access is granted. r: The user has paid the subscription fee. Express the following system specifications using the propositions p, q, and r and logical connectives (including negations). Access is granted whenever the user has paid the subscription fee and enters a valid password.

(r ∧ p) → q

Let p, q, r be the propositions: p: The user enters a valid password. q: Access is granted. r: The user has paid the subscription fee. Express the following system specifications using the propositions p, q, and r and logical connectives (including negations). If the user has not entered a valid password but has paid the subscription fee, then access is granted.

(¬p ∧ r) → q

An explorer is captured by a group of cannibals. There are two types of cannibals—those who always tell the truth and those who always lie. The cannibals will barbecue the explorer unless he can determine whether a particular cannibal always lies or always tells the truth. He is allowed to ask the cannibal exactly one question. Why does the question "Are you a liar?" not work?

Both types of cannibals will answer "no." Explanation If the explorer encounters a truth-teller, then he will honestly answer "no" to her question. If she encounters a liar, then the honest answer to her question is "yes," so he will lie and answer "no." Thus everybody will answer "no" to the question, and the explorer will have no way to determine which type of cannibal she is speaking to.

Four friends have been identified as suspects for an unauthorized access into a computer system. They have made statements to the investigating authorities. Alice said, "Carlos did it." John said, "I did not do it." Carlos said, "Diana did it." Diana said, "Carlos lied when he said that I did it." If the authorities know that exactly one is lying, who did it?

Carlos did it. Explanation Since Carlos and Diana are making contradictory statements, the liar must be one of them. Therefore, Alice is telling the truth, so Carlos did it. John and Diana are telling the truth as well here, and it is Carlos who is lying.

Consider the following specifications: "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." The goal of this exercise is to determine whether the given specifications are consistent. To do this, we express the given specifications in terms of the statements: u: The software system is being upgraded.a: Users can access the file system.s: Users can save new files. Identify a set of truth values that make the three specification true.

FFT, TFT, FFF, and FTT

Four friends have been identified as suspects for an unauthorized access into a computer system. They have made statements to the investigating authorities. Alice said, "Carlos did it." John said, "I did not do it." Carlos said, "Diana did it." Diana said, "Carlos lied when he said that I did it." If the authorities know that exactly one of the four suspects is telling the truth, who did it?

John did it. Explanation John did it. There are four cases to consider: If Alice is the sole truth-teller, then Carlos did it, but this means that John is telling the truth, a contradiction. If John is the sole truth-teller, then Diana must be lying, so she did it, but then Carlos is telling the truth, a contradiction. If Carlos is the sole truth-teller, then Diana did it, but that makes John truthful, again a contradiction. So, the only possibility is that Diana is the sole truth-teller. This means that John is lying when he denied it, so he did it.

An explorer is captured by a group of cannibals. There are two types of cannibals—those who always tell the truth and those who always lie. The cannibals will barbecue the explorer unless he can determine whether a particular cannibal always lies or always tells the truth. He is allowed to ask the cannibal exactly one question. Which of the following questions helps in determining whether the cannibal she is speaking to is a truth-teller or a liar? Select all the questions that work. (Check all that apply.)

The questions that help in determining whether the cannibal she is speaking to is a truth-teller or a liar are as follows: Is the color of the sky blue? If I were to ask you if you always told the truth, would you say that you did? Explanation There are several possible correct answers. One is the following question: "If I were to ask you if you always told the truth, would you say that you did?" Then if the cannibal is a truth teller, he will answer yes (truthfully), while if he is a liar, then, since in fact he would have said that he did tell the truth if questioned, he will now lie and answer no.

Consider the following specifications: "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." The goal of this exercise is to determine whether the given specifications are consistent. To do this, we express the given specifications in terms of the statements: u: The software system is being upgraded.a: Users can access the file system.s: Users can save new files. What is the symbolic representation of the statement, "If users cannot save new files, then the system software is not being upgraded"?

The symbolic representation of the given statement is ¬s → ¬u.

Consider the following specifications: "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." The goal of this exercise is to determine whether the given specifications are consistent. To do this, we express the given specifications in terms of the statements: u: The software system is being upgraded.a: Users can access the file system.s: Users can save new files. Is the given system consistent?

Yes, the system is consistent. Explanation System specifications are consistent when they do not contain conflicting requirements to derive a contradiction. When specifications are not consistent, there is no way to develop a system that satisfies all specifications. So, the given system is consistent.

Consider the following specifications: "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." The goal of this exercise is to determine whether the given specifications are consistent. To do this, we express the given specifications in terms of the statements: u: The software system is being upgraded.a: Users can access the file system.s: Users can save new files. What is the symbolic representation of the statement, "If users can access the file system, then they can save new files"?

a → s

Let p, q, r be the propositions: p: The user enters a valid password. q: Access is granted. r: The user has paid the subscription fee. Express the following system specifications using the propositions p, q, and r and logical connectives (including negations). The user has paid the subscription fee, but does not enter a valid password.

r ∧ ¬p

Consider the following specifications: "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." The goal of this exercise is to determine whether the given specifications are consistent. To do this, we express the given specifications in terms of the statements: u: The software system is being upgraded.a: Users can access the file system.s: Users can save new files. What is the symbolic representation of the statement, "Whenever the system software is being upgraded, users cannot access the file system"?

u → ¬a

Let p, q, r be the propositions: p: The user enters a valid password. q: Access is granted. r: The user has paid the subscription fee. Express the following system specifications using the propositions p, q, and r and logical connectives (including negations). Access is denied if the user has not paid the subscription fee.

¬r → ¬q