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Chapter 1: Problem 28

Translate each of these statements into logical expressions using predicates, quantifiers, and logical connectives. a) Something is not in the correct place. b) All tools are in the correct place and are in excellent condition. c) Everything is in the correct place and in excellent condition. d) Nothing is in the correct place and is in excellent condition. e) One of your tools is not in the correct place, but it is in excellent condition.

Short Answer

Expert verified
a) \( \exists x eg P(x) \); b) \( \forall x (P(x) \land Q(x)) \); c) \( \forall x (P(x) \land Q(x)) \); d) \( \forall x eg (P(x) \land Q(x)) \); e) \( \exists x (eg P(x) \land Q(x))) \).

Step by step solution

01

Define Predicates

Let P(x) represent 'x is in the correct place' and Q(x) represent 'x is in excellent condition.'
02

Translate statement (a)

Something is not in the correct place. This translates to the existential quantifier with negation: \( \exists x eg P(x) \).
03

Translate statement (b)

All tools are in the correct place and are in excellent condition. This translates to the universal quantifier: \( \forall x (P(x) \land Q(x)) \).
04

Translate statement (c)

Everything is in the correct place and in excellent condition. This is also a universal quantifier similar to (b): \( \forall x (P(x) \land Q(x)) \).
05

Translate statement (d)

Nothing is in the correct place and is in excellent condition. This translates to the universal quantifier with negation: \( \forall x eg (P(x) \land Q(x)) \).
06

Translate statement (e)

One of your tools is not in the correct place, but it is in excellent condition. This is represented with an existential quantifier that includes both conditions: \( \exists x (eg P(x) \land Q(x)) \).

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Key Concepts

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

Understanding Predicates
Predicates are statements that contain variables, and they become propositions when the variables are specified. For example, if we have a predicate P(x) that represents 'x is in the correct place', P(x) will be either true or false depending on whether x is indeed in the correct place. Predicates allow us to make general statements about objects in a specific domain.
Using predicates, we can construct logical expressions that can be universally or existentially quantified, allowing us to discuss complex logical scenarios efficiently.
What Are Quantifiers?
Quantifiers are tools used in logic to specify the extent to which a predicate is true over a range of elements. There are two main types of quantifiers: the universal quantifier and the existential quantifier.
The universal quantifier, symbolized as \( \forall \), means 'for all' or 'for every'. It indicates that the predicate holds true for all elements in a given domain.
Conversely, the existential quantifier, symbolized as \( \exists \), means 'there exists'. It indicates that there is at least one element in the domain for which the predicate holds true.
Logical Connectives
Logical connectives are symbols that connect two or more logical statements to form a compound statement. Some common logical connectives include:
  • \( eg \) (not): Negates the truth value of a statement.
  • \( \rightarrow \) (implies): Indicates that one statement implies another.
  • \( \beta \) (and): Means both statements must be true.
  • \( \beta \) (or): Means at least one of the statements must be true.
In the context of our exercise, using these connectives helps construct precise logical expressions that capture the intended meaning of the statements.
Existential Quantifier in Detail
The existential quantifier \( \exists \) is used when we want to assert that at least one element in the domain of discourse satisfies a given predicate. For example, if we say 'Something is not in the correct place', we can express this in logical form as \( \exists x eg P(x) \). This means there is at least one x for which the predicate P(x) is false.
The existential quantifier is crucial when we want to indicate that a condition applies to at least one, but not necessarily all, members of the domain.
Universal Quantifier Explained
The universal quantifier \( \forall \) indicates that a predicate holds true for every element in the domain. For instance, the statement 'All tools are in the correct place and are in excellent condition' can be translated into logical notation as \( \forall x (P(x) \land Q(x)) \). This means for every x, both P(x) and Q(x) are true.
Universal quantifiers are essential in logic to express statements that apply universally within a certain context. They ensure that the predicate's condition must be true for every element, offering a comprehensive assertion.

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Most popular questions from this chapter

For each of these arguments determine whether the argument is correct or incorrect and explain why.a) Everyone enrolled in the university has lived in a dormitory. Mia has never lived in a dormitory. Therefore, Mia is not enrolled in the university. b) A convertible car is fun to drive. Isaac鈥檚 car is not a convertible. Therefore, Isaac鈥檚 car is not fun to drive. c) Quincy likes all action movies. Quincy likes the movie Eight Men Out. Therefore, Eight Men Out is an action movie. d) All lobstermen set at least a dozen traps. Hamilton is a lobsterman. Therefore, Hamilton sets at least a dozen traps.

Find a counterexample, if possible, to these universally quantified statements, where the domain for all variables consists of all integers. a) \(\forall x \forall y\left(x^{2}=y^{2} \rightarrow x=y\right)\) b) \(\forall x \exists y\left(y^{2}=x\right)\) c) \(\forall x \forall y(x y \geq x)\)

Exercises \(48-51\) establish rules for null quantification that we can use when a quantified variable does not appear in part of a statement. Establish these logical equivalences, where \(x\) does not occur as a free variable in \(A\) . Assume that the domain is nonempty. $$ \begin{array}{l}{\text { a) } \forall x(P(x) \rightarrow A) \equiv \exists x P(x) \rightarrow A} \\ {\text { b) } \exists x(P(x) \rightarrow A) \equiv \forall x P(x) \rightarrow A}\end{array} $$

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

For each of these arguments, explain which rules of inference are used for each step. a) 鈥淟inda, a student in this class, owns a red convertible. Everyone who owns a red convertible has gotten at least one speeding ticket. Therefore, someone in this class has gotten a speeding ticket.鈥� b) 鈥淓ach of five roommates, Melissa, Aaron, Ralph, Veneesha, and Keeshawn, has taken a course in discrete mathematics. Every student who has taken a course in discrete mathematics can take a course in algorithms. Therefore, all five roommates can take a course in algorithms next year.鈥� c) 鈥淎ll movies produced by John Sayles are wonder-ful. John Sayles produced a movie about coal miners. Therefore, there is a wonderful movie about coal miners.鈥� d) 鈥淭here is someone in this class who has been to France. Everyone who goes to France visits the Louvre. Therefore, someone in this class has visited the Louvre.鈥�
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