Mathematical Logic Excercises, Exercises of Mathematical logic

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MATHEMATICAL LOGIC EXERCISES

Chiara Ghidini and Luciano Serafini

Anno Accademico 2013-

We thank Annapaola Marconi for her work in previous editions of this booklet.

Contents

• 1 Introduction
• 2 Propositional Logic
  • 2.1 Basic Concepts
  • 2.2 Truth Tables
  • 2.3 Propositional Formalization
    • 2.3.1 Formalizing Simple Sentences
    • 2.3.2 Formalizing Problems
  • 2.4 Normal Form Reduction
• 3 First Order Logic
  • 3.1 Basic Concepts
  • 3.2 FOL Formalization
• 4 Modal Logic
  • 4.1 Basic Concepts
  • 4.2 Satisfiability and Validity
  • 4.3 Modal Logic Formalization

Mathematics is the only instructional material that can be presented in an entirely undogmatic way. The Mathematical Intelligencer, v. 5, no. 2, 1983

Chapter 1 MAX DEHN

Introduction

The purpose of this booklet is to give you a number of exercises on proposi- tional, first order and modal logics to complement the topics and exercises covered during the lectures of the course on mathematical logic. The mate- rial presented here is not a direct component of the course but is offered to you as an incentive and a support to understand and master the concepts and exercises presented during the course.

Symbol Difficulty  Trivial  Easy  Medium  Difficult  Very difficult

Propositional Logic

Exercise 2.2.  - Let’s consider the interpretation v where v(p) = F, v(q) = T, v(r) = T. Does v satisfy the following propositional formulas?

1. (p → ¬q) ∨ ¬(r ∧ q)
2. (¬p ∨ ¬q) → (p ∨ ¬r)
3. ¬(¬p → ¬q) ∧ r
4. ¬(¬p → q ∧ ¬r)

Solution.

v satisfies 1., 3. and 4. v doesn’t satisfy 2.

2.2 Truth Tables

Exercise 2.3.  - Compute the truth table of (F ∨ G) ∧ ¬(F ∧ G).

Solution.

F G F ∨ G F ∧ G ¬(F ∧ G) (F ∨ G) ∧ ¬(F ∧ G) T T T T F F T F T F T T F T T F T T F F F F T F

• The formula models an exclusive or!

]

2.2 Truth Tables

Exercise 2.4.  - Use the truth tables method to determine whether (p → q) ∨ (p → ¬q) is valid.

Solution.

p q p → q ¬q p → ¬q (p → q) ∨ (p → ¬q) T T T F F T T F F T T T F T T F T T F F T T T T The formula is valid since it is satisfied by every interpretation.

]

Exercise 2.5.  - Use the truth tables method to determine whether (¬p∨q)∧(q → ¬r∧¬p)∧(p∨r) (denoted with ϕ) is satisfiable.

Solution.

p q r ¬p ∨ q ¬r ∧ ¬p q → ¬r ∧ ¬p (p ∨ r) ϕ T T T T F F T F T T F T F F T F T F T F F T T F T F F F F T T F F T T T F F T F F T F T T T F F F F T T F T T T F F F T T T F F There exists an interpretation satisfying ϕ, thus ϕ is satisfiable.

]

2.2 Truth Tables

• (p → p) → p
• p → (p → p)
• p ∨ q → p ∧ q
• p ∨ (q ∧ r) → (p ∧ r) ∨ q
• p → (q → p)
• (p ∧ ¬q) ∨ ¬(p ↔ q)

]

Exercise 2.9.  Use the truth table method to verify whether the following formulas are valid, satisfiable or unsatisfiable:

• (p → q) ∧ ¬q → ¬p
• (p → q) → (p → ¬q)
• (p ∨ q → r) ∨ p ∨ q
• (p ∨ q) ∧ (p → r ∧ q) ∧ (q → ¬r ∧ p)
• (p → (q → r)) → ((p → q) → (p → r))
• (p ∨ q) ∧ (¬q ∧ ¬p)
• (¬p → q) ∨ ((p ∧ ¬r) ↔ q)
• (p → q) ∧ (p → ¬q)
• (p → (q ∨ r)) ∨ (r → ¬p)

]

Propositional Logic

Exercise 2.10.  Use the truth table method to verify whether the following logical consequences and equivalences are correct:

• (p → q) |= ¬p → ¬q
• (p → q) ∧ ¬q |= ¬p
• p → q ∧ r |= (p → q) → r
• p ∨ (¬q ∧ r) |= q ∨ ¬r → p
• ¬(p ∧ q) ≡ ¬p ∨ ¬q
• (p ∨ q) ∧ (¬p → ¬q) ≡ q
• (p ∧ q) ∨ r ≡ (p → ¬q) → r
• (p ∨ q) ∧ (¬p → ¬q) ≡ p
• ((p → q) → q) → q ≡ p → q

2.3 Propositional Formalization

2.3.1 Formalizing Simple Sentences

Exercise 2.11.  - Let’s consider a propositional language where

• p means “Paola is happy”,
• q means “Paola paints a picture”,
• r means “Renzo is happy”.

Formalize the following sentences:

Propositional Logic

Exercise 2.13.  - Let A =“Aldo is Italian” and B =“Bob is English”. Formalize the following sentences:

1. “Aldo isn’t Italian”
2. “Aldo is Italian while Bob is English”
3. “If Aldo is Italian then Bob is not English”
4. “Aldo is Italian or if Aldo isn’t Italian then Bob is English”
5. “Either Aldo is Italian and Bob is English, or neither Aldo is Italian nor Bob is English”

Solution.

1. ¬A
2. A ∧ B
3. A → ¬B
4. A ∨ (¬A → B) logically equivalent to A ∨ B
5. (A ∧ B) ∨ (¬A ∧ ¬B) logically equivalent to A ↔ B

]

Exercise 2.14.  Angelo, Bruno and Carlo are three students that took the Logic exam. Let’s consider a propositional language where

• A =“Aldo passed the exam”,
• B =“Bruno passed the exam”,
• C =“Carlo passed the exam”.

Formalize the following sentences:

2.3 Propositional Formalization

1. “Carlo is the only one passing the exam”
2. “Aldo is the only one not passing the exam”
3. “Only one, among Aldo, Bruno and Carlo, passed the exam”
4. “At least one among Aldo, Bruno and Carlo passed”
5. “At least two among Aldo, Bruno and Carlo passed the exam”
6. “At most two among Aldo, Bruno and Carlo passed the exam”
7. “Exactly two, among Aldo, Bruno and Carlo passed the exam”

]

Exercise 2.15.  - Let’s consider a propositional langiage where

• A =“Angelo comes to the party”,
• B =“Bruno comes to the party”,
• C =“Carlo comes to the party”,
• D =“Davide comes to the party”.

Formalize the following sentences:

1. “If Davide comes to the party then Bruno and Carlo come too”
2. “Carlo comes to the party only if Angelo and Bruno do not come”
3. “Davide comes to the party if and only if Carlo comes and Angelo doesn’t come”
4. “If Davide comes to the party, then, if Carlo doesn’t come then Angelo comes”
5. “Carlo comes to the party provided that Davide doesn’t come, but, if Davide comes, then Bruno doesn’t come”

2.3 Propositional Formalization

3. “If Angelo and Bruno come to the party, then Carlo comes provided that Davide doesn’t come”
4. “Carlo comes to the party if Bruno and Angelo don’t come, or if Davide comes”
5. “If Angelo comes to the party then Bruno or Carlo come too, but if Angelo doesn’t come to the party, then Carlo and Davide come”

]

Exercise 2.17.  - Socrate says:

“If I’m guilty, I must be punished; I’m guilty. Thus I must be punished.”

Is the argument logically correct?

Solution. The argument is logically correct: if p means “I’m guilty” and q means “I must be punished”, then: (p → q) ∧ p |= q (modus ponens)

]

Exercise 2.18.  - Socrate says:

“If I’m guilty, I must be punished; I’m not guilty. Thus I must not be punished.”

Is the argument logically correct?

Propositional Logic

Solution. The argument is not logically correct: (p → q) ∧ ¬p 2 ¬q

• consider for instance v(p) = F and v(q) = T

]

Exercise 2.19.  Socrate says:

“If I’m guilty, I must be punished; I must not be punished. Thus I’m not guilty.”

Is the argument logically correct?

]

Exercise 2.20.  Socrate says:

“If I’m guilty, I must be punished; I must be punished. Thus I’m guilty.”

Is the argument logically correct?

]

Exercise 2.21.  Formalize the following arguments and verify whether they are correct:

• “If Carlo won the competition, then either Mario came second or Sergio came third. Sergio didn’t come third. Thus, if Mario didn’t come second, then Carlo didn’t win the competition.”