Homework problems: Nested quantifiers(1.9-1.10)

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

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

1. ∀x∀y∃z (z = (x - y)/3) (If false, give acounterexample.)

1. ∀x∀y (xy = yx) (If false, give acounterexample.)

1. ∃x∃y∃z (x2 + y2 = z2) (If true, give anexample.)

1. Redo the above (problem 1), with the domain of positiveintegers.

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1. Translate each of the following English statements intological expressions. The domain of discourse is the set of allintegers.

   1. There are two numbers whose sum is equal to theirproduct.
      

1. The product of every two positive integers ispositive.
   

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

1. There is a positive integer that is smaller than allother positive integers.

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

   1. No one has ever beat Nancy.

1. Everyone has been beaten before.

1. Everyone has won at least one game.

1. No one has beaten both Ingrid and Dominic.

1. There are two members who have never beenbeaten.

1. Translate each of the following English statements intological expressions. The domain of discourse is the set of all realnumbers.

   1. The reciprocal of every positive number ispositive.

1. There is no smallest number.

1. There are two numbers whose ratio is less than1.

1. Write the negation of each of the following logicalexpressions so that all negations immediately precedepredicates.

   1. ∀x ∃y ∃z P(y, x, z)

1. ∃x ∃y P(x, y) ∧ ∀x ∀y Q(x, y)

1. ∃x ∀y ( P(x, y) ↔ P(y, x) )

1. ∃x ∀y ( P(x, y) → Q(x, y) )

Homework problems: Logical reasoning(1.11-1.13)

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

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

   • I will either drive on the freeway or take surfacestreets.

   • 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

1. Use the laws of logic to prove the conclusion from thehypotheses. Give propositions and predicate variable names in yourproof. Use the set of all students as the domain of discourse. Thehypotheses are:

   • Larry and Hubert are taking Boolean Logic.

   • Any student who takes Boolean Logic can takeAlgorithms.

Conclude that Larry and Hubert can takeAlgorithms.

1. Use the laws of logic to prove the conclusion from thehypotheses. Give propositions and predicate variable names in yourproof. Use the set of all people as the domain of discourse. Thehypotheses are:

   • Everyone who practices hard is a goodmusician.

   • There is a member of the orchestra who practiceshard.

Conclude that someone in the orchestra is a goodmusician.

1. Which of the following arguments are valid? Explain yourreasoning.

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

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

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