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Chapter 1: Problem 48

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

Short Answer

Expert verified
(a) \( (\forall x P(x)) \vee A \equiv \forall x (P(x) \vee A) \)(b) \( (\exists x P(x)) \vee A \equiv \exists x (P(x) \vee A) \)

Step by step solution

01

Understand the Statement

The exercise asks to prove two logical equivalences where a quantified variable does not appear in part of a statement, assuming the domain is nonempty.
02

Analyze Case (a)

We need to prove \( (\forall x P(x)) \vee A \equiv \forall x (P(x) \vee A) \), assuming that \( x \) does not occur as a free variable in \( A \).
03

Expand the Meaning of Quantifiers

For any arbitrary domain element, \( \forall x P(x) \) means \( P(x) \) is true for all \( x \). \( \exists x P(x) \) means there is some \( x \) for which \( P(x) \) is true.
04

Consider the Logical OR (\( \vee \)) Operator

For any statement \( A \), either \( A \) is true or false. The disjunction \( (\forall x P(x)) \vee A \) means \( A \) can be true independently of \( P(x) \).
05

Prove (a) Using Substitution

We need to show that both \( (\forall x P(x)) \vee A \) and \( \forall x (P(x) \vee A) \) result in the same truth values.If \( A \) is true, both sides are true regardless of \( P(x) \).If \( A \) is false, then \( P(x) \) must be true for all \( x \). Hence both \( (\forall x P(x)) \vee A \) and \( \forall x (P(x) \vee A) \) have the same truth value.
06

Analyze Case (b)

We need to prove \( (\exists x P(x)) \vee A \equiv \exists x(P(x) \vee A) \), assuming that \( x \) does not occur as a free variable in \( A \).
07

Prove (b) Using Substitution

We need to show that both \( (\exists x P(x)) \vee A \) and \( \exists x (P(x) \vee A) \) result in the same truth values.If \( A \) is true, both sides are true.If \( A \) is false, then there must exist some \( x \) for which \( P(x) \) is true. Hence both \( (\exists x P(x)) \vee A \) and \( \exists x (P(x) \vee A) \) have the same truth value.

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Key Concepts

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

Quantifiers
In mathematical logic, quantifiers help us make statements about quantities or amounts. The two main types of quantifiers are the universal quantifier, \(\forall\), and the existential quantifier, \(\there exists x\). The universal quantifier states that a given condition is true for all elements in a particular set. On the other hand, the existential quantifier states that there is at least one element in a set for which the condition holds true. Understanding these quantifiers is essential for working with logical statements and proofs.
Logical OR
The logical OR operator, represented by \(\vee\), joins two statements and asserts that at least one of them is true. In the context of quantification, \((\forall x P(x)) \vee A\) means that the statement P(x) holds for all x or statement A is true, or both. The operator ensures flexibility in logical statements by accommodating multiple conditions. If either condition is satisfied, the entire expression evaluates to true. This operator is crucial for combining statements and exploring different possibilities in logical scenarios.
Free Variables
A free variable is a variable that is not bound by a quantifier within a statement. For example, in \(A = [B(x) \wedge \forall y P(y)]\), \(x\) is a free variable. When dealing with quantifiers, it's important to note whether variables are free or bound. Bound variables are specified within the scope of a quantifier, while free variables are not. Statements involving both should be carefully analyzed to ensure their validity and meaningful interpretation.
Truth Values
Truth values refer to the evaluation of logical statements as true or false. In logical proofs, we examine statements under different conditions to determine their truth values. For instance, the expression \((\forall x P(x)) \vee A\) will always be true if either \(A\) is true or \(P(x)\) holds for all \(x\). Evaluating the truth values of logical expressions helps to establish equivalences, such as the ones in the given exercise. This examination ensures logical consistency and soundness in mathematical proofs.
Mathematical Logic
Mathematical logic is the field of mathematics that deals with formal logical systems and their applications. It includes the study of various logical operators, quantifiers, and their interactions in forming complex statements. In mathematical logic, we use precise language to construct arguments, prove theorems, and derive valid conclusions. The given exercise utilizes mathematical logic to establish equivalences regarding quantified statements, demonstrating how logical operators work together to produce meaningful, truthful expressions.

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Most popular questions from this chapter

Exercises \(61-64\) are based on questions found in the book Symbolic Logic by Lewis Carroll. Let P(x), Q(x), R(x), and S(x) be the statements 鈥渪 is a baby,鈥� 鈥渪 is logical,鈥� 鈥渪 is able to manage a crocodile,鈥� and 鈥渪 is despised,鈥� respectively. Suppose that the domain consists of all people. Express each of these statements using quantifiers; logical connectives; and P(x), Q(x), R(x), and S(x). a) Babies are illogical. b) Nobody is despised who can manage a crocodile. c) Illogical persons are despised. d) Babies cannot manage crocodiles. e) Does (d) follow from (a), (b), and (c)? If not, is there a correct conclusion?

Let \(Q(x)\) be the statement " \(x+1>2 x\) . If the domain consists of all integers, what are these truth values? $$ \begin{array}{llll}{\text { a) }} & {Q(0)} & {\text { b) } Q(-1)} & {\text { c) }} \quad {Q(1)} \\ {\text { d) }} & {\exists x Q(x)} & {\text { e) } \quad \forall x Q(x)} & {\text { f) } \quad \exists x \neg Q(x)}\end{array} $$ g) \(\quad \forall x \neg Q(x)\)

Suppose that the domain of the propositional function \(P(x)\) consists of the integers \(0,1,2,3,\) and \(4 .\) Write out each of these propositions using disjunctions, conjunctions, and negations. $$ \begin{array}{llll}{\text { a) }} & {\exists x P(x)} & {\text { b) } \forall x P(x)} & {\text { c) }} \quad {\exists x \neg P(x)} \\ {\text { d) }} & {\forall x \neg P(x)} & {\text { e) } \neg \exists x P(x)} & {\text { f) } \neg \forall x P(x)}\end{array} $$

Prove or disprove that you can use dominoes to tile the standard checkerboard with two adjacent corners removed (that is, corners that are not opposite).

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.
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