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First-Order Predicate Logic

Description: This quiz covers the fundamentals of First-Order Predicate Logic, including syntax, semantics, and inference rules.
Number of Questions: 15
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Tags: first-order predicate logic propositional logic fol logic
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Which of the following is a valid syntax for a first-order predicate logic statement?

  1. ∀x(Px → Qx)

  2. ∃x(Px ∧ Qx)

  3. (Px ∨ Qx) → ∀x(Px ∨ Qx)

  4. None of the above

Show answer
Correct Option: A
Explanation:

The correct syntax for a first-order predicate logic statement is ∀x(Px → Qx), which represents "For all x, if x has property P, then x has property Q".

What is the meaning of the predicate symbol 'P(x)' in first-order predicate logic?

  1. It represents a property that can be true or false for an object x.

  2. It represents a set of objects that satisfy a certain condition.

  3. It represents a function that maps an object x to a truth value.

  4. None of the above

Show answer
Correct Option: A
Explanation:

In first-order predicate logic, a predicate symbol P(x) represents a property that can be true or false for an object x.

Which of the following is a valid inference rule in first-order predicate logic?

  1. Modus ponens

  2. Modus tollens

  3. Hypothetical syllogism

  4. All of the above

Show answer
Correct Option: D
Explanation:

Modus ponens, modus tollens, and hypothetical syllogism are all valid inference rules in first-order predicate logic.

What is the difference between a universal quantifier (∀) and an existential quantifier (∃) in first-order predicate logic?

  1. A universal quantifier asserts that a property holds for all objects in a domain, while an existential quantifier asserts that it holds for at least one object.

  2. A universal quantifier asserts that a property holds for some objects in a domain, while an existential quantifier asserts that it holds for all objects.

  3. A universal quantifier asserts that a property holds for no objects in a domain, while an existential quantifier asserts that it holds for some objects.

  4. None of the above

Show answer
Correct Option: A
Explanation:

A universal quantifier (∀) asserts that a property holds for all objects in a domain, while an existential quantifier (∃) asserts that it holds for at least one object.

What is the negation of the statement "∀x(Px → Qx)" in first-order predicate logic?

  1. ∃x(Px ∧ ¬Qx)

  2. ∃x(¬Px ∨ Qx)

  3. ∀x(¬Px ∨ Qx)

  4. None of the above

Show answer
Correct Option: A
Explanation:

The negation of the statement "∀x(Px → Qx)" is "∃x(Px ∧ ¬Qx)", which represents "There exists an object x such that x has property P and x does not have property Q".

Which of the following is a valid first-order predicate logic statement?

  1. ∀x(Px → Qx) ∧ ∃x(¬Px)

  2. ∀x(Px → Qx) → ∃x(¬Qx)

  3. ∃x(Px ∧ Qx) → ∀x(Px ∨ Qx)

  4. None of the above

Show answer
Correct Option: A
Explanation:

The statement "∀x(Px → Qx) ∧ ∃x(¬Px)" is a valid first-order predicate logic statement because it represents "For all x, if x has property P, then x has property Q, and there exists an object x that does not have property P".

What is the domain of discourse in first-order predicate logic?

  1. The set of all objects under consideration.

  2. The set of all properties under consideration.

  3. The set of all statements under consideration.

  4. None of the above

Show answer
Correct Option: A
Explanation:

The domain of discourse in first-order predicate logic is the set of all objects under consideration.

Which of the following is a valid first-order predicate logic statement?

  1. ∀x(Px ∨ Qx) → (∀xPx ∨ ∀xQx)

  2. ∃x(Px ∧ Qx) → (∃xPx ∧ ∃xQx)

  3. ∀x(Px → Qx) → (∃xPx → ∃xQx)

  4. None of the above

Show answer
Correct Option: A
Explanation:

The statement "∀x(Px ∨ Qx) → (∀xPx ∨ ∀xQx)" is a valid first-order predicate logic statement because it represents "For all x, if x has property P or x has property Q, then either all x have property P or all x have property Q".

What is the difference between a constant symbol and a variable symbol in first-order predicate logic?

  1. A constant symbol represents a specific object, while a variable symbol represents any object in the domain of discourse.

  2. A constant symbol represents any object in the domain of discourse, while a variable symbol represents a specific object.

  3. A constant symbol represents a property, while a variable symbol represents an object.

  4. None of the above

Show answer
Correct Option: A
Explanation:

In first-order predicate logic, a constant symbol represents a specific object, while a variable symbol represents any object in the domain of discourse.

Which of the following is a valid first-order predicate logic statement?

  1. ∀x(Px → Qx) → (∃xPx → ∃xQx)

  2. ∃x(Px ∧ Qx) → (∃xPx ∨ ∃xQx)

  3. ∀x(Px ∨ Qx) → (∃xPx ∧ ∃xQx)

  4. None of the above

Show answer
Correct Option: A
Explanation:

The statement "∀x(Px → Qx) → (∃xPx → ∃xQx)" is a valid first-order predicate logic statement because it represents "For all x, if x has property P, then x has property Q. Therefore, if there exists an object x that has property P, then there exists an object x that has property Q".

What is the difference between a term and a formula in first-order predicate logic?

  1. A term is a constant symbol, a variable symbol, or a function symbol applied to terms, while a formula is a statement that can be true or false.

  2. A term is a statement that can be true or false, while a formula is a constant symbol, a variable symbol, or a function symbol applied to terms.

  3. A term is a constant symbol or a variable symbol, while a formula is a function symbol applied to terms.

  4. None of the above

Show answer
Correct Option: A
Explanation:

In first-order predicate logic, a term is a constant symbol, a variable symbol, or a function symbol applied to terms, while a formula is a statement that can be true or false.

Which of the following is a valid first-order predicate logic statement?

  1. ∀x(Px → Qx) → (∀xPx → ∀xQx)

  2. ∃x(Px ∧ Qx) → (∃xPx ∧ ∃xQx)

  3. ∀x(Px ∨ Qx) → (∀xPx ∨ ∃xQx)

  4. None of the above

Show answer
Correct Option: A
Explanation:

The statement "∀x(Px → Qx) → (∀xPx → ∀xQx)" is a valid first-order predicate logic statement because it represents "For all x, if x has property P, then x has property Q. Therefore, if all x have property P, then all x have property Q".

What is the difference between a closed formula and an open formula in first-order predicate logic?

  1. A closed formula contains no free variables, while an open formula contains at least one free variable.

  2. A closed formula contains at least one free variable, while an open formula contains no free variables.

  3. A closed formula is a statement that can be true or false, while an open formula is a term.

  4. None of the above

Show answer
Correct Option: A
Explanation:

In first-order predicate logic, a closed formula contains no free variables, while an open formula contains at least one free variable.

Which of the following is a valid first-order predicate logic statement?

  1. ∀x(Px ∨ Qx) → (∀xPx ∨ ∃xQx)

  2. ∃x(Px ∧ Qx) → (∃xPx ∨ ∀xQx)

  3. ∀x(Px → Qx) → (∃xPx ∨ ∀xQx)

  4. None of the above

Show answer
Correct Option: A
Explanation:

The statement "∀x(Px ∨ Qx) → (∀xPx ∨ ∃xQx)" is a valid first-order predicate logic statement because it represents "For all x, if x has property P or x has property Q, then either all x have property P or there exists an object x that has property Q".

What is the difference between a model and a structure in first-order predicate logic?

  1. A model is a set of objects and a set of relations on those objects, while a structure is a set of objects, a set of relations on those objects, and a set of functions on those objects.

  2. A model is a set of objects, a set of relations on those objects, and a set of functions on those objects, while a structure is a set of objects and a set of relations on those objects.

  3. A model is a set of objects and a set of functions on those objects, while a structure is a set of objects, a set of relations on those objects, and a set of functions on those objects.

  4. None of the above

Show answer
Correct Option: A
Explanation:

In first-order predicate logic, a model is a set of objects and a set of relations on those objects, while a structure is a set of objects, a set of relations on those objects, and a set of functions on those objects.

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