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Chapter 1: Problem 34

Express the negation of these propositions using quantifiers, and then express the negation in English. a) Some drivers do not obey the speed limit. b) All Swedish movies are serious. c) No one can keep a secret. d) There is someone in this class who does not have a good attitude.

Short Answer

Expert verified
a) Every driver obeys the speed limit. b) There is at least one Swedish movie that is not serious. c) There is at least one person who can keep a secret. d) Everyone in this class has a good attitude.

Step by step solution

01

Title - Identify Original Proposition and Formulate The Quantifier

Identify the proposition and write the original statement using quantifiers. a) \textquote{Some drivers do not obey the speed limit} can be written as: \(\exists x \, (D(x) \land eg O(x))\) b) \textquote{All Swedish movies are serious} can be written as: \(\forall x \, (S(x) \rightarrow Srs(x))\) c) \textquote{No one can keep a secret} can be written as: \(\forall x \, (P(x) \rightarrow eg Ks(x))\) d) \textquote{There is someone in this class who does not have a good attitude} can be written as: \(\exists x \, (C(x) \land eg Ga(x))\)
02

Title - Apply De Morgan's Laws to Negate Each Statement

For each quantified statement, apply De Morgan's laws to find the negation. a) The negation of \(\exists x \, (D(x) \land eg O(x))\) is \(\forall x \, eg (D(x) \land eg O(x))\), which simplifies to: \(\forall x \, (eg D(x) \lor O(x))\) b) The negation of \(\forall x \, (S(x) \rightarrow Srs(x))\) is \(\exists x \, eg (S(x) \rightarrow Srs(x))\), which simplifies to: \(\exists x \, (S(x) \land eg Srs(x))\) c) The negation of \(\forall x \, (P(x) \rightarrow eg Ks(x))\) is \(\exists x \, eg (P(x) \rightarrow eg Ks(x))\), which simplifies to: \(\exists x \, (P(x) \land Ks(x))\) d) The negation of \(\exists x \, (C(x) \land eg Ga(x))\) is \(\forall x \, eg (C(x) \land eg Ga(x))\), which simplifies to: \(\forall x \, (eg C(x) \lor Ga(x))\)
03

Title - Convert Negated Statements to English

Translate the negated quantifier expressions back into English. a) \(\forall x \, (eg D(x) \lor O(x))\) translates to: \textquote{Every driver obeys the speed limit.} b) \(\exists x \, (S(x) \land eg Srs(x))\) translates to: \textquote{There is at least one Swedish movie that is not serious.} c) \(\exists x \, (P(x) \land Ks(x))\) translates to: \textquote{There is at least one person who can keep a secret.} d) \(\forall x \, (eg C(x) \lor Ga(x))\) translates to: \textquote{Everyone in this class has a good attitude.}

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Key Concepts

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

Negation
Negation in logic is the process of determining the opposite of a given statement. In simpler terms, if a proposition is true, then its negation is false, and vice versa. The symbol for negation is usually \( eg \). For example, if the original statement is 'Some drivers do not obey the speed limit,' its negation would be 'Every driver obeys the speed limit.' To negate quantified statements, we often use logical rules and transformations, such as De Morgan's laws, which we'll cover in another section.
De Morgan's Laws
De Morgan's Laws are critical tools in understanding and manipulating logical statements, especially when dealing with negations of quantified sentences. These laws show us how to distribute negations inside logical expressions. The two basic rules are:
  • The negation of a conjunction is the disjunction of the negations: \( eg (A \land B) \equiv (eg A \lor eg B) \)
  • The negation of a disjunction is the conjunction of the negations: \( eg (A \lor B) \equiv (eg A \land eg B) \)
For instance, when negating 'Some drivers do not obey the speed limit,' we would transform \( eg \exists x (D(x) \land eg O(x)) \) using De Morgan's laws to \( \forall x (eg D(x) \lor O(x)) \).
Predicate Logic
Predicate logic extends propositional logic by dealing with predicates and quantifiers. Predicates are statements or functions that can return true or false values depending on their arguments. For example, in 'All Swedish movies are serious,' \( S(x) \) could represent 'x is a Swedish movie' and \( Srs(x) \) could represent 'x is serious.' Predicate logic also employs two main quantifiers: the universal quantifier (\( \forall \)) and the existential quantifier (\( \exists \)). These quantify the statements over a domain of discourse, allowing us to create more complex and expressive logical formulas.
Quantifiers
Quantifiers are symbols used in logic to express the extent to which a predicate applies to a set of elements. The two primary quantifiers are:
  • Universal Quantifier (\( \forall \)): States that a predicate holds for all elements in a domain. For example, \( \forall x \ (S(x) \rightarrow Srs(x)) \) means 'For all x, if x is a Swedish movie, then x is serious.'
  • Existential Quantifier (\( \exists \)): Indicates that there is at least one element in the domain for which the predicate is true. For example, \( \exists x \ (D(x) \land eg O(x)) \) means 'There exists at least one x such that x is a driver and x does not obey the speed limit.'
Using quantifiers, we can formulate precise and comprehensive logical statements. This precision is particularly helpful in mathematical proofs and reasoning.

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Most popular questions from this chapter

Use rules of inference to show that if \(\forall x(P(x) \vee Q(x))\) \(\forall x(\neg Q(x) \vee S(x)), \quad \forall x(R(x) \rightarrow \neg S(x)),\) and \(\exists x \neg P(x)\) are true, then \(\exists x \neg R(x)\) is true.

Use predicates, quantifiers, logical connectives, and mathematical operators to express the statement that every positive integer is the sum of the squares of four integers.

Suppose that the domain of the propositional function \(P(x)\) consists of the integers \(1,2,3,4,\) and \(5 .\) Express these statements without using quantifiers, instead using only negations, disjunctions, and conjunctions. $$ \begin{array}{ll}{\text { a) } \quad \exists x P(x)} & {\text { b) } \forall x P(x)} \\ {\text { c) } \quad \neg \exists x P(x)} & {\text { d) } \neg \forall x P(x)}\end{array} $$ e) \(\quad \forall x((x \neq 3) \rightarrow P(x)) \vee \exists x \neg P(x)\)

Fuzzy logic is used in artificial intelligence. In fuzzy logic, a proposition has a truth value that is a number between 0 and 1, inclusive. A proposition with a truth value of 0 is false and one with a truth value of 1 is true. Truth values that are between 0 and 1 indicate varying degrees of truth. For instance, the truth value 0.8 can be assigned to the statement 鈥淔red is happy,鈥� because Fred is happy most of the time, and the truth value 0.4 can be assigned to the statement 鈥淛ohn is happy,鈥� because John is happy slightly less than half the time. Use these truth values to solve The truth value of the negation of a proposition in fuzzy logic is 1 minus the truth value of the proposition. What are the truth values of the statements 鈥淔red is not happy鈥� and 鈥淛ohn is not happy鈥�?

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.
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