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Let S(x) be the predicate 鈥渪 is a student,鈥� F(x) the predicate 鈥渪 is a faculty member,鈥� and A(x, y) the predicate 鈥渪 has asked y a question,鈥� where the domain consists of all people associated with your school. Use quantifiers to express each of these statements. a) Lois has asked Professor Michaels a question. b) Every student has asked Professor Gross a question. c) Every faculty member has either asked Professor Miller a question or been asked a question by Professor Miller. d) Some student has not asked any faculty member a question. e) There is a faculty member who has never been asked a question by a student. f ) Some student has asked every faculty member a question. g) There is a faculty member who has asked every other faculty member a question. h) Some student has never been asked a question by a faculty member.

Step by step solution

01

Translate the predicates

Identify and translate each of the predicates into quantifiers. The predicates are as follows: S(x) means 'x is a student,' F(x) means 'x is a faculty member,' and A(x, y) means 'x has asked y a question.'

02

Lois has asked Professor Michaels a question

Lois and Professor Michaels are specific individuals. Let L denote Lois and M denote Professor Michaels. The statement 'Lois has asked Professor Michaels a question' can be written as: \(A(L, M)\)

03

Every student has asked Professor Gross a question

Professor Gross is a specific individual, so let G represent Professor Gross. 'Every student has asked Professor Gross a question' can be expressed using universal quantification: \(\forall x (S(x) \rightarrow A(x, G))\)

04

Every faculty member has either asked Professor Miller a question or been asked a question by Professor Miller

Professor Miller is a specific individual, denoted as M. The statement can be expressed as: \(\forall x (F(x) \rightarrow (A(x, M) \lor A(M, x)))\)

05

Some student has not asked any faculty member a question

This involves both existential and universal quantifiers. The specific statement can be expressed as: \(\exists x (S(x) \wedge \forall y (F(y) \rightarrow \eg A(x, y)))\)

06

There is a faculty member who has never been asked a question by a student

Represent this using existential and universal quantifiers: \(\exists x (F(x) \wedge \forall y (S(y) \rightarrow \eg A(y, x)))\)

07

Some student has asked every faculty member a question

Use a combination of existential and universal quantifiers to represent this statement: \(\exists x (S(x) \wedge \forall y (F(y) \rightarrow A(x, y)))\)

08

There is a faculty member who has asked every other faculty member a question

This statement can be written as: \(\exists x (F(x) \wedge \forall y (F(y) \rightarrow (x \eq y \rightarrow A(x, y))))\)

09

Some student has never been asked a question by a faculty member

This final statement is expressed as: \(\exists x (S(x) \wedge \forall y (F(y) \rightarrow \eg A(y, x)))\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quantifiers

Quantifiers are essential in predicate logic to denote the scope of a statement over a domain of discourse. There are two primary types of quantifiers: universal quantifiers and existential quantifiers.

Universal quantifiers (denoted by \(\forall\)) indicate that a statement holds for all elements within the domain. For example, \(\forall x P(x)\) translates to 'P(x) is true for all x in the domain.'

Existential quantifiers (denoted by \(\exists\)) indicate that a statement holds for at least one element within the domain. For example, \(\exists x P(x)\) means 'There exists at least one x in the domain for which P(x) is true.'

• Universal Quantifier: \(\forall x S(x)\)
• Existential Quantifier: \(\exists x S(x)\)

Understanding the appropriate use of these quantifiers is crucial for accurately expressing logical statements involving different subjects and objects in predicate logic.

Logical Expressions

In predicate logic, logical expressions combine predicates and quantifiers to form statements about the domain. Logical expressions use logical connectives like AND (\(\wedge\)), OR (\(\lor\)), and implication (\(\rightarrow\)).

For example, if S(x) means 'x is a student' and A(x, y) means 'x has asked y a question':

• \

Universal Quantification

Universal quantification is a type of quantifier that asserts a predicate holds true for all elements within the specified domain. It is denoted by \(\forall\). For example, the universal quantification \(\forall x (S(x) \rightarrow A(x, y))\) means 'For all x, if x is a student, then x has asked y a question.'

In predicate logic, universal quantification is often used to create generalized statements about a set of objects. Here are a few key points about universal quantification:

• Universal quantification requires that every element in the domain satisfies the predicate.
• It is essential in defining properties or behaviors that apply universally within a given context.

Using universal quantification ensures that the logical statements holistically account for all possible subjects, thus making the statements more robust and generalized. In the given exercise, \(\forall x (S(x) \rightarrow A(x, G))\) conveys that every student has asked Professor Gross a question.

Existential Quantification

Existential quantification is another key concept in predicate logic that expresses the existence of at least one element in the domain for which a predicate is true. This is denoted by \(\exists\).

For example, \(\exists x (S(x) \wedge \forall y (F(y) \rightarrow \eg A(x, y)))\) means 'There exists at least one x such that x is a student and for all y, if y is a faculty member, then x has not asked y a question.'

• Existential quantification focuses on the existence of one or more elements satisfying the predicate.
• It is used to highlight specific instances or to prove the presence of particular characteristics or behaviors within a domain.

In the given exercise, \(\exists x (F(x) \wedge \forall y (S(y) \rightarrow \eg A(y, x)))\) translates to 'There is a faculty member who has never been asked a question by a student,' emphasizing the existence of at least one such individual.