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Chapter 1: Q3E (page 53)

LetQ(x,y) denote the statement “ xis the capaital of y” what are these truth values?

(a)Q (Denver,Colorado)

(b)Q (Detroit,Michigan)

(c)Q (Massachusetts,Boston)

(d) Q (Newyork,Newyork)

Short Answer

Expert verified

The truth values of Q(Denver,Colorado),Q(Detroit,Michigan),Q(Massachusetts,Boston)and Q(Newyork,Newyork) are true, false, false, and false respectively. For this, check the capital in a given statements and determine the truth values based on this.

Step by step solution

01

Definition of the truth value  

A truth valuein a truth table indicates the truth (T or 1) or falsity (F or 0) of a specified proposition or statement.

02

To find the truth values of given propositions.

(a)Q (Denver,Colorado)

The given condition is “xis the capital ofy” .

The given proposition isQ (Denver,Colorado).

Here, the Denver is the capital of Colorado.

Therefore, the statementQ (Denver,Colorado)is true.

(b)Q (Detroit,Michigan)

The given condition is “xis the capital ofy” .

The given proposition isQ (Detroit,Michigan).

Here, the Detroit is not the capital of Colorado.

Therefore, the statementQ (Detroit,colorado)is false.

(c)Q (Massachusetts,Boston)

The given condition is “xis the capital ofy” .

The given proposition isQ (Massachusetts,Boston).

Here, the Boston is the capital of Massachusetts.

Therefore, the statementQ (Massachusetts,Boston)is false.

(d)Q (Newyork,Newyork)

The given condition is “xis the capital ofy” .

The given proposition isQ (Newyork,Newyork).

Here, the New York is not the capital of New York.

Therefore, the statementQ (Newyork,Newyork)is false.

The truth values ofQ (Denver,Colorado),

Q (Detroit,Michigan),Q (Massachusetts,Boston)
and Q (Newyork,Newyork) are true, false, false, and false respectively.

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Most popular questions from this chapter

Translate these statements into English, where the domain for each variable consists of all real numbers.

(a)∀x∃y(x<y)

(b)∀x∀y(x≥0∧y≥0→xy≥0)

(c)∀x∀y∃z(xy=z)

Suppose that during the most recent fiscal year, the annual revenue of Acme Computer was billion dollars and its net profit was billion dollars, the annual revenue of Nadir Software was billion dollars and its net profit was billion dollars, and the annual revenue of Quixote Media was billion dollars and its net profit was billion dollars. Determine the truth value of each of these propositions for the most recent fiscal year.

  1. Quixote Media had the largest annual revenue.
  2. Nadir Software had the lowest net profit and Acme Computer had the largest annual revenue.
  3. Acme Computer had the largest net profit or Quixote Media had the largest net profit.
  4. If Quixote Media had the smallest net profit, then Acme Computer had the largest annual revenue.
  5. Nadir Software had the smallest net profit if and only if Acme Computer had the largest annual revenue.

Show that(p∨q)∧(¬p∨r)→(q∨r) is a tautology

A says “We are both knaves” and B says nothing. Exercises 24–31 relate to inhabitants of an island on which there are three kinds of people: knights who always tell the truth, knaves who always lie, and spies (called normals by Sullying [Sm78]) who can either lie or tell the truth. You encounter three people, A, B, and C. You know one of these people is a knight, one is a knave, and one is a spy. Each of the three people knows the type of person each of other two is. For each of these situations, if possible, determine whether there is a unique solution and determine who the knave, knight, and spy are. When there is no unique solution, list all possible solutions or state that there are no solutions

A says “I am not the spy,” B says “I am not the spy,” and C says “A is the spy.”

A says “We are both knaves” and B says nothing. Exercises 24–31 relate to inhabitants of an island on which there are three kinds of people: knights who always tell the truth, knaves who always lie, and spies (called normals by Sullying [Sm78]) who can either lie or tell the truth. You encounter three people, A, B, and C. You know one of these people is a knight, one is a knave, and one is a spy. Each of the three people knows the type of person each of other two is. For each of these situations, if possible, determine whether there is a unique solution and determine who the knave, knight, and spy are. When there is no unique solution, list all possible solutions or state that there are no solutions

A says “I am the knight,” B says “I am the knight,” and C says “I am the knight.”
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