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Chapter 1: Problem 51

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. $$ \begin{array}{l}{\text { a) } \forall x(P(x) \rightarrow A) \equiv \exists x P(x) \rightarrow A} \\ {\text { b) } \exists x(P(x) \rightarrow A) \equiv \forall x P(x) \rightarrow A}\end{array} $$

Short Answer

Expert verified
Both (a) \( \forall x (P(x) \rightarrow A) \rightarrow \text{A} \) and (b) are true due to the domain indifference between x and non-impact on A.

Step by step solution

01

Analyze the Logical Equivalence for (a)

Given: \ \( \forall x(P(x) \rightarrow A) \text{ is equivalent to } \ \exists x P(x) \rightarrow A \) \ \ To show this is true, we need to recognize that if the domain is non-empty and no free variable \(x\) appears in \(A\), then \(A\)'s truth value is independent of \(P(x)\). Hence we can establish the equivalence by showing the contexts where the implications hold true.
02

Implication Analysis

Both expressions \( \forall x(P(x) \rightarrow A) \) and \( \exists x P(x) \rightarrow A \) indicate when \(A\) should hold true. Notice: \( P(x) \rightarrow A \) simplifies to \( eg P(x) \lor A \) which means, for all \(x\), \(P(x) \rightarrow A\) is equivalent to \(A\) if \(P(x)\) is false for all x, or if \(A\) holds true by itself.
03

Establishing Equivalence

We see that \( \forall x (P(x) \rightarrow A) \) says that \(A\) holds regardless of \(P(x)\). Similarly, \( \exists x P(x) \rightarrow A \) also implies that even if \(P(x) \) holds for some \(x\), \(A\) must be true. Given that \(A\) does not involve \(x\), both expressions yield the same result, establishing their equivalence.
04

Analyze the Logical Equivalence for (b)

Given: \ \( \exists x (P(x) \rightarrow A) \text{ is equivalent to } \forall x P(x) \rightarrow A \) \ \ Following the same notion that \(A\) is independent of \(x\), if there's at least one \(x\) in the domain where \(P(x) \rightarrow A\) happens, based on the non-empty domain, equivalences need to be validated similarly.
05

Simplifying (b)

Analogous to the logic in (a), if \(P(x) \rightarrow A\) works for some \(x\), then \(A\) holds. So, if there鈥檚 some \(x\) for the implication, \( \forall x P(x) \rightarrow A \) should be valid overall & vice versa,
06

Conclusion

Both \( \exists x (P(x) \rightarrow A) \) ensures \(A\)'s validity under existing \(x\). Confirming the consistency when subjected to, \( \forall x (P(x) \rightarrow A)\). Thus equivalence for (a) and (b) follows.

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Key Concepts

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

Quantification
Quantification is a fundamental concept in logic that allows us to make general statements about elements within a domain. The two main types are universal and existential quantification.

Universal quantification, denoted as \(\forall x\), states that a proposition holds for all elements in a domain. For instance, \(\forall x P(x)\) asserts that property \(\text{P}\) is true for every element \(\text{x}\) in the domain.

Existential quantification, denoted as \(\forall x \), claims that there exists at least one element in the domain for which the proposition is true. For example, \(\forall x P(x)\) means there is at least one \(\text{x}\) such that \(\text{P}\) is valid.

Understanding these symbols and their meanings is critical when analyzing logical equivalences, like those presented in the exercise.
Free Variable
A free variable is a variable that is not bound by a quantifier within a logical expression. It can take any value within the context of the expression.

Consider the formula \(\forall x (P(x) \rightarrow A)\), in which \(\text{A}\) does not include the variable \(\text{x}\). Here, \(\text{x}\) inside \(\text{P(x)}\) is bound by the quantifier \(\forall x\), making it a bound variable. However, \(\text{A}\) is independent of \(\text{x}\), so any occurrences of \(\text{x}\) outside \(\text{P(x)}\) would be free.

Correct identification of free and bound variables is crucial for determining the logical equivalence of statements, as it affects how you interpret the scope and applicability of propositions.
Logical Implication
Logical implication, denoted by the arrow symbol \(\rightarrow \), is a relationship between two statements where if the first statement (the antecedent) is true, then the second statement (the consequent) must also be true.

For example, \(P(x) \rightarrow A\) means if \(\text{P(x)}\) is true, then \(\text{A}\) must be true. If \(\text{P(x)}\) is false, the implication still holds true regardless of \(\text{A}\)'s validity. This aspect can simplify logical expressions by converting them into equivalent forms, such as \(eg P(x) \lor \ A\) (read as 'not \(\text{P}\)' or '\text{A}').

Logical implications help in understanding the relationships within quantified statements. In the given exercises, we used this concept to demonstrate equivalencies by restructuring the implications and proving their consistencies against different scenarios in the domain.

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Most popular questions from this chapter

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 disjunction of two propositions in fuzzy logic is the maximum of the truth values of the two propositions. What are the truth values of the statements 鈥淔red is happy, or John is happy鈥� and 鈥淔red is not happy, or John is not happy鈥�?

Express the negations of these propositions using quantifiers, and in English. a) Every student in this class likes mathematics. b) There is a student in this class who has never seen a computer. c) There is a student in this class who has taken every mathematics course offered at this school. d) There is a student in this class who has been in at least one room of every building on campus.

A statement is in prenex normal form (PNF) if and only if it is of the form $$ Q_{1} x_{1} Q_{2} x_{2} \cdots Q_{k} x_{k} P\left(x_{1}, x_{2}, \ldots, x_{k}\right) $$ where each \(Q_{i}, i=1,2, \ldots, k,\) is either the existential quantifier or the universal quantifier, and \(P\left(x_{1}, \ldots, x_{k}\right)\) is a predicate involving no quantifiers. For example, \(\exists x \forall y(P(x, y) \wedge Q(y))\) is in prenex normal form, whereas \(\exists x P(x) \vee \forall x Q(x)\) is not (because the quantifiers do not all occur first). Every statement formed from propositional variables, predicates, \(\mathbf{T},\) and \(\mathbf{F}\) using logical connectives and quantifiers is equivalent to a statement in prenex normal form. Exercise 51 asks for a proof of this fact. Show how to transform an arbitrary statement to a statement in prenex normal form that is equivalent to the given statement. (Note: A formal solution of this exercise requires use of structural induction, covered in Section \(5.3 . )\)

Let \(C(x)\) be the statement " \(x\) has a cat," let \(D(x)\) be the statement " \(x\) has a dog," and let \(F(x)\) be the statement "x has a ferret." Express each of these statements in terms of \(C(x), D(x), F(x),\) quantifiers, and logical connectives. Let the domain consist of all students in your class. a) A student in your class has a cat, a dog, and a ferret. b) All students in your class have a cat, a dog, or a ferret. c) Some student in your class has a cat and a ferret, but not a dog. d) No student in your class has a cat, a dog, and a ferret. e) For each of the three animals, cats, dogs, and ferrets, there is a student in your class who has this animal as a pet.

Translate these statements into English, where \(C(x)\) is " \(x\) is a comedian" and \(F(x)\) is " \(x\) is funny" and the domain consists of all people. $$ \begin{array}{ll}{\text { a) } \forall x(C(x) \rightarrow F(x))} & {\text { b) } \forall x(C(x) \wedge F(x))} \\ {\text { c) } \quad \exists x(C(x) \rightarrow F(x))} & {\text { d) } \exists x(C(x) \wedge F(x))}\end{array} $$
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