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Chapter 1: Problem 3

Let \(Q(x, y)\) be the statement "x has sent an e-mail message to \(y,\) " where the domain for both \(x\) and \(y\) consists of all students in your class. Express each of these quantifications in English. $$ \begin{array}{ll}{\text { a) } \exists x \exists y Q(x, y)} & {\text { b) } \exists x \forall y Q(x, y)} \\ {\text { c) } \forall x \exists y Q(x, y)} & {\text { d) } \exists y \forall x Q(x, y)} \\ {\text { e) } \forall y \exists x Q(x, y)} & {\text { f) } \forall x \forall y Q(x, y)}\end{array} $$

Short Answer

Expert verified
a) At least one student has sent an e-mail to another student. b) One student has e-mailed everyone. c) Every student has sent an e-mail to at least one student. d) There is one student everyone has e-mailed. e) For each student, someone has sent them an e-mail. f) Every student has e-mailed every other student.

Step by step solution

01

Understanding the Problem

You are given a predicate logic statement about students sending e-mails to each other. The task is to translate each quantified statement into English.
02

Step a) Translating \( \exists x \exists y Q(x, y) \)

\( \exists x \exists y Q(x, y) \) means there exists at least one student \( x \) and at least one student \( y \) such that \( x \) has sent an e-mail message to \( y \).
03

Step b) Translating \( \exists x \forall y Q(x, y) \)

\( \exists x \forall y Q(x, y) \) means there exists at least one student \( x \) such that \( x \) has sent an e-mail message to every student \( y \).
04

Step c) Translating \( \forall x \exists y Q(x, y) \)

\( \forall x \exists y Q(x, y) \) means for every student \( x \), there exists at least one student \( y \) such that \( x \) has sent an e-mail message to \( y \).
05

Step d) Translating \( \exists y \forall x Q(x, y) \)

\( \exists y \forall x Q(x, y) \) means there exists at least one student \( y \) such that every student \( x \) has sent an e-mail message to \( y \).
06

Step e) Translating \( \forall y \exists x Q(x, y) \)

\( \forall y \exists x Q(x, y) \) means for every student \( y \), there exists at least one student \( x \) such that \( x \) has sent an e-mail message to \( y \).
07

Step f) Translating \( \forall x \forall y Q(x, y) \)

\( \forall x \forall y Q(x, y) \) means every student \( x \) has sent an e-mail message to every student \( y \).

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Key Concepts

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

Quantified Statements
Quantified statements are powerful tools in predicate logic. They allow us to make assertions about elements within a domain. The two primary types are universal and existential quantifiers. Universal quantifiers assert something about every element in a domain, while existential quantifiers assert that there is at least one element in the domain for which the predicate holds.
Translation
Translating quantified statements into natural language is an essential skill in understanding predicate logic. For instance, given a predicate Q(x, y) which means 'x has sent an e-mail to y,' we can translate various quantified statements into clear English sentences. Each translation helps in comprehending how the statements reflect relationships within the provided domain.
Existential Quantifier
The existential quantifier, denoted by \(\backslash ex x\), states that 'there exists' at least one element such that a given predicate is true. For example: \(\backslash ex x \backslash ex y Q(x, y)\) means 'there exists at least one student x and at least one student y such that x has sent an e-mail to y.' This captures the idea that the action of sending an e-mail has occurred at least once within the group.
Universal Quantifier
The universal quantifier, denoted by \(\backslash forall x \), states that 'for all' elements in a domain, a predicate holds true. For instance, \(\backslash forall x \backslash ex y Q(x, y)\) is translated as 'for every student x, there exists at least one student y such that x has sent an e-mail to y.' This indicates every student has sent an e-mail to at least one other student.
Predicate
A predicate is a statement that contains variables and becomes a proposition when the variables are specified. In our example, Q(x, y) is a predicate meaning 'x has sent an e-mail to y.' Understanding predicates helps in forming complex logical statements and determining their truth values based on the elements within their domains.

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Most popular questions from this chapter

Let M(x, y) be 鈥渪 has sent y an e-mail message鈥� and T(x, y) be 鈥渪 has telephoned y,鈥� where the domain consists of all students in your class. Use quantifiers to express each of these statements. (Assume that all e-mail messages that were sent are received, which is not the way things often work.) a) Chou has never sent an e-mail message to Koko. b) Arlene has never sent an e-mail message to or tele- phoned Sarah. c) Jose has never received an e-mail message from Deborah. d) Every student in your class has sent an e-mail mes- sage to Ken. e) No one in your class has telephoned Nina. f ) Everyone in your class has either telephoned Avi or sent him an e-mail message. g) There is a student in your class who has sent every- one else in your class an e-mail message. h) There is someone in your class who has either sent an e-mail message or telephoned everyone else in your class. i) There are two different students in your class who have sent each other e-mail messages. j) There is a student who has sent himself or herself an e-mail message. k) There is a student in your class who has not received an e-mail message from anyone else in the class and who has not been called by any other student in the class. l) Every student in the class has either received an email message or received a telephone call from another student in the class. m) There are at least two students in your class such that one student has sent the other e-mail and the second student has telephoned the first student. n) There are two different students in your class who between them have sent an e-mail message to or telephoned everyone else in the class.

Express each of these statements using logical operators, predicates, and quantifiers. a) Some propositions are tautologies. b) The negation of a contradiction is a tautology. c) The disjunction of two contingencies can be a tautology. d) The conjunction of two tautologies is a tautology.

Write out \(\exists ! x P(x),\) where the domain consists of the integers \(1,2,\) and \(3,\) in terms of negations, conjunctions, and disjunctions.

Determine whether these are valid arguments. a) If \(x\) is a positive real number, then \(x^{2}\) is a positive real number. Therefore, if \(a^{2}\) is positive, where \(a\) is a real number, then \(a\) is a positive real number. b) If \(x^{2} \neq 0,\) where \(x\) is a real number, then \(x \neq 0 .\) Let \(a\) be a real number with \(a^{2} \neq 0 ;\) then \(a \neq 0\)

Find a counterexample, if possible, to these universally quantified statements, where the domain for all variables consists of all real numbers. $$ \begin{array}{ll}{\text { a) } \forall x\left(x^{2} \neq x\right)} & {\text { b) } \forall x\left(x^{2} \neq 2\right)} \\ {\text { c) } \forall x(|x|>0)} \end{array} $$
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