Question

Homework problems: Nested quantifiers (1.9-1.10)

1. Determine the truth value of each expression below if the domain is the set of all real numbers.

   1. ∃x∀y (xy = 0) (If true, give an example.)

2. ∀x∀y∃z (z = (x - y)/3) (If false, give a counterexample.)

3. ∀x∀y (xy = yx) (If false, give a counterexample.)

4. ∃x∃y∃z (x2 + y2 = z2) (If true, give an example.)

2. Redo the above (problem 1), with the domain of positive integers.

3.   

3. Translate each of the following English statements into logical expressions. The domain of discourse is the set of all integers.

   1. There are two numbers whose sum is equal to their product.
      

2. The product of every two positive integers is positive.
   

3. Every positive integer can be expressed as the sum of the squares of four integers.

4. There is a positive integer that is smaller than all other positive integers.

4. The domain of discourse is the members of a chess club. The predicate B(x, y) means that person x has beaten person y at some point in time. Give a logical expression equivalent to the following English statements.

   1. No one has ever beat Nancy.

2. Everyone has been beaten before.

3. Everyone has won at least one game.

4. No one has beaten both Ingrid and Dominic.

5. There are two members who have never been beaten.

5. Translate each of the following English statements into logical expressions. The domain of discourse is the set of all real numbers.

   1. The reciprocal of every positive number is positive.

2. There is no smallest number.

3. There are two numbers whose ratio is less than 1.

6. Write the negation of each of the following logical expressions so that all negations immediately precede predicates.

   1. ∀x ∃y ∃z P(y, x, z)

2. ∃x ∃y P(x, y) ∧ ∀x ∀y Q(x, y)

3. ∃x ∀y ( P(x, y) ↔ P(y, x) )

4. ∃x ∀y ( P(x, y) → Q(x, y) )

Homework problems: Logical reasoning (1.11-1.13)

7. Use a truth table to prove the conclusion from the hypotheses. The hypotheses are:

   • If I drive on the freeway, I will see the fire.

   • I will either drive on the freeway or take surface streets.

   • I am not going to take surface streets.

Conclude that I will see the fire.

Use the following variable names:

• p: I drive on the freeway

• r: I take surface streets

• q: I see the fire

p	q	r

8. Use the laws of logic to prove the conclusion from the hypotheses. Give propositions and predicate variable names in your proof. Use the set of all students as the domain of discourse. The hypotheses are:

   • Larry and Hubert are taking Boolean Logic.

   • Any student who takes Boolean Logic can take Algorithms.

Conclude that Larry and Hubert can take Algorithms.

9. Use the laws of logic to prove the conclusion from the hypotheses. Give propositions and predicate variable names in your proof. Use the set of all people as the domain of discourse. The hypotheses are:

   • Everyone who practices hard is a good musician.

   • There is a member of the orchestra who practices hard.

Conclude that someone in the orchestra is a good musician.

10. Which of the following arguments are valid? Explain your reasoning.

    • I have a student in my class who is getting an A. Therefore, John, a student in my class is getting an A.
      
      
      
      

    • Every girl scout who sells at least 50 boxes of cookies will get a prize. Suzy, a girl scout, got a prize. Therefore Suzy sold 50 boxes of cookies.

11. Use the laws of logic to show that ∀x(P(x) ∧ Q(x)) implies that ∀x Q(x) ∧ ∀x P(x).

Answer 1