Hwk#1-1.1-1.2

In context of the inhabitants of the island of knights and knaves created by Smullyan, where knights always tell the truth and knaves always lie. You encounter two people, A and B. Determine what A and B are if they address you as: A says "The two of us are both knights" and B says "A is a knave."

A is a knave and B is a knight.

For # 32-33. 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. Explain why the question "Are you a liar?" does not work.

Both types of cannibals will answer with "no".

"4 + x = 5" is a proposition, because it is a declarative sentence.

False

If P = "2 + 1 = 3", then the negation of P is "2 × 1 ≠ 3".

False

Let p and q be the propositions p: It is below freezing. q: It is snowing. The compound proposition for the statement "It is either snowing or below freezing (or both)" is p ∧ q.

False

Which of the following questions does work in determining whether the cannibal she is speaking to is a truth teller or a liar? Select all the questions that work.

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?

Identify all the negations of the proposition "Mei has an MP3 player".

It is not the case that Mei has an MP3 player. & Mei has no MP3 player.

"What time is it ?" is a proposition.

No

Let P: There is no pollution in New Jersey. The statement "The whole world is polluted" is the negation of P. Is the above statement true?

No

Let p and q be the propositions p: "Swimming at the New Jersey shore is allowed." q: "Sharks have been spotted near the shore." Identify the compound proposition p ∧ q.

Swimming at the New Jersey shore is allowed, and sharks have been spotted near the shore.

Let p and q be the propositions p: "Swimming at the New Jersey shore is allowed." q: "Sharks have been spotted near the shore." Identify the compound proposition ¬p ∨ q as an English sentence.

Swimming at the New Jersey shore is not allowed, or sharks have been spotted near the shore.

Let p and q be the propositions p : It is below freezing. q : It is snowing. Identify the correct steps that are used to find the compound proposition for the statement "It is not below freezing and it is not snowing."

The proposition for the statement "It is not below freezing" is ¬p. & The proposition for the statement "It is not snowing" is ¬q. & The proposition for the statement "It is not below freezing and it is not snowing" is ¬p ∧ ¬q.

Let P: The summer in Maine is hot and sunny. The negation of P: _____.

The summer in Maine is not hot or it is not sunny.

Identify the true statement about the sentence "The moon is made of green cheese."

This is a proposition that is false.

Which of the following statements are true about the sentence "There are no black flies in Maine"?

This is a proposition. & Its truth value is false, as anyone who has been to Maine knows.

Identify the correct statements about "2n ≥ 100."

This is not a proposition. & Its truth value depends on the value of n. & Its truth value depends on the value of 2n.

Which of the following is true for the statement "Do not pass go"?

This is not a proposition; it's a command.

Let p and q be the propositions p : It is below freezing. q : It is snowing. The compound proposition for the statement "It is below freezing and snowing" is p ∧ q.

True

Let p and q be the propositions p: "Swimming at the New Jersey shore is allowed." q: "Sharks have been spotted near the shore." Check if the proposition, "Sharks have not been spotted near the shore" is the expression for ¬q.

True

Is the given system consistent?

Yes

Let p, q, and r be the propositions p: You have the flu. q: You miss the final examination. r: You pass the course. Identify the statement that express the compound proposition p ∨ q ∨ r as an English sentence.

You have the flu, or miss the final exam, or pass the course.

What is the symbolic representation of the statement, "If users can access the file system, then they can save new files."?

a → s

Determine whether the "or" in the given statement is an inclusive "or" or an exclusive "or": Lunch includes soup or salad.

exclusive "or"

Determine whether the "or" in the given statement is an inclusive "or" or an exclusive "or": Publish or perish.

exclusive "or"

Determine whether the "or" in the given statement is an inclusive "or" or an exclusive "or": Experience with C++ or Java is required.

inclusive "or"

Determine whether the "or" in the given statement is an inclusive "or" or an exclusive "or": To enter the country you need a passport or a voter registration card.

inclusive "or"

Let p and q be the propositions p: It is below freezing. q: It is snowing. The compound proposition for the statement "It is below freezing but not snowing." is _____.

p ∧ ¬q

Let p and q be the propositions p: You drive over 65 miles per hour. q: You get a speeding ticket. Identify the expression that represents the proposition "You drive over 65 miles per hour, but you do not get a speeding ticket" using p and q and logical connectives (including negations).

p ∧ ¬q

Let p, q, and r be the propositions p :You get an A on the final exam. q :You do every exercise in this book. r :You get an A in this class. Identify the expression for the proposition "You get an A on the final, but you don't do every exercise in this book; nevertheless, you get an A in this class." using p, q, and r and logical connectives (including negations).

p ∧ ¬q ∧ r

Let p and q be the propositions p: You drive over 65 miles per hour. q: You get a speeding ticket. Identify the expression that represents the proposition "You get a speeding ticket, but you do not drive over 65 miles per hour" using p and q and logical connectives (including negations).

q ∧ ¬p

Let p, q, and r be the propositions p :You get an A on the final exam. q :You do every exercise in this book. r :You get an A in this class. Identify the expression for the proposition "To get an A in this class, it is necessary for you to get an A on the final." using p, q, and r and logical connectives (including negations).

r → p

For #28-31. Consider these 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 these specifications are consistent. To do this, we express these 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 and q be the propositions p: You drive over 65 miles per hour. q: You get a speeding ticket. Identify the expression that represents the proposition "You do not drive over 65 miles per hour" using p and q and logical connectives (including negations).

¬p

What is the symbolic representation of the statement, "If users cannot save new files, then the system software is not being upgraded."?

¬s → ¬u