Homework Week 2

Problem 15

What Google search would you use to look for Web pages relating to Ethiopian restaurants in New York or New Jersey?

Problem 17

Suppose that in Example 7, the inscriptions on Trunks 1, 2, and 3 are:

Trunk 1: “The treasure is in Trunk 3.”
Trunk 2: “The treasure is in Trunk 1.”
Trunk 3: “This trunk is empty.”

For each of these statements, determine whether the Queen who never lies could state this, and if so, which trunk the treasure is in.

1. “All the inscriptions are false.”
   
2. “Exactly one of the inscriptions is true.”
   
3. “Exactly two of the inscriptions are true.”
   
4. “All three inscriptions are true.”

Problem 21

When three professors are seated in a restaurant, the hostess asks them: “Does everyone want coffee?” The first professor says “I do not know.” The second professor then says “I do not know.” Finally, the third professor says “No, not everyone wants coffee.” The hostess comes back and gives coffee to the professors who want it. How did she figure out who wanted coffee?

Problem 23

A says “At least one of us is a knave” and B says nothing.

Exercises 23-27 relate to inhabitants of the island of knights and knaves, where knights always tell the truth and knaves always lie. You encounter two people, A and B. Determine, if possible, what A and B are if they address you in the ways described. If you cannot determine what these two people are, can you draw any conclusions?

Problem 25

Exact text not pasted in chat. Listed in the official Week 2 assignment for Section 1.2.

Problem 27

A says “We are both knaves” and B says nothing.

Problem 29

Exact text not pasted in chat. Listed in the official Week 2 assignment for Section 1.2.

Problem 31

Exact text not pasted in chat. Listed in the official Week 2 assignment for Section 1.2.

Problem 37

Exact text not pasted in chat. Listed in the official Week 2 assignment for Section 1.2.

Read tables 6, 7, and 8.

Problem 1

Use truth tables to verify these equivalences.

1. \(p \land T \equiv p\)
   
2. \(p \lor F \equiv p\)
   
3. \(p \land F \equiv F\)
   
4. \(p \lor T \equiv T\)
   
5. \(p \lor p \equiv p\)
   
6. \(p \land p \equiv p\)

Problems 3, 5, 7

Exact text not pasted in chat. Listed in the official Week 2 assignment for Section 1.3.

Problem 11

Show that each of these conditional statements is a tautology by using truth tables.

1. \((p \land q) \to p\)
   
2. \(p \to (p \lor q)\)
   
3. \(\neg p \to (p \to q)\)
   
4. \((p \land q) \to (p \to q)\)
   
5. \(\neg(p \to q) \to p\)
   
6. \(\neg(p \to q) \to \neg q\)

Problems 21 and 23

Problem 21 text pasted below. Problem 23 exact text not pasted in chat.

Problem 21

Show that \(\neg(p \leftrightarrow q)\) and \(p \leftrightarrow \neg q\) are logically equivalent.

Problem 33

Exact text not pasted in chat. Listed in the official Week 2 assignment for Section 1.3.

Problem 45

Find a compound proposition involving the propositional variables \(p\), \(q\), and \(r\) that is true when exactly two of \(p\), \(q\), and \(r\) are true and is false otherwise.

Hint: Form a disjunction of conjunctions. Include a conjunction for each combination of values for which the compound proposition is true. Each conjunction should include each of the three propositional variables or its negations.