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Chapter 1: Problem 19

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) \(\quad \forall x((x \neq 3) \rightarrow P(x)) \vee \exists x \neg P(x)\)

Short Answer

Expert verified
Combine logical expressions about each element of the domain using conjunctions and disjunctions.

Step by step solution

01

Understand the Problem

Given a propositional function \(P(x)\) with a domain consisting of the integers 1, 2, 3, 4, and 5, the goal is to express the given logical statements without using quantifiers, instead using only negations, disjunctions, and conjunctions.
02

Express \( \forall x P(x) \)

The statement \( \forall x P(x) \) means that \(P(x)\) is true for all values in the domain. This can be written as: \[P(1) \text{ AND } P(2) \text{ AND } P(3) \text{ AND } P(4) \text{ AND } P(5).\]
03

Express \( eg \forall x P(x) \)

The negation of \( \forall x P(x) \) means that it is not true that \(P(x)\) is true for all values. This can be written as the negation of the conjunction: \[eg (P(1) \text{ AND } P(2) \text{ AND } P(3) \text{ AND } P(4) \text{ AND } P(5)).\]
04

Express \( eg \forall x P(x) \)

The statement \( eg \forall x P(x) \) means that there exists an x for which \(P(x)\) is not true. This can be written as: \[ eg P(1) \text{ OR } eg P(2) \text{ OR } eg P(3) \text{ OR } eg P(4) \text{ OR } eg P(5).\]
05

Express \( eg \forall x P(x) \)

The given proposition is \[ \forall x((x eq 3) \rightarrow P(x)) \text{ OR } eg P(x).\] Rephrasing the first part without quantifiers, for every x, if \(x\) is not equal to 3, then \(P(x)\) and combine it with the disjunction: \[(1 eq 3 \rightarrow P(1)) \text{ AND } (2 eq 3 \rightarrow P(2)) \text{ AND } (3 eq 3 \rightarrow P(3)) \text{ AND } (4 eq 3 \rightarrow P(4)) \text{ AND } (5 eq 3 \rightarrow P(5)) \text{ OR } P(1) \text{ OR } P(2) \text{ OR } P(3) \text{ OR } P(4) \text{ AND } (P(5)\]

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Key Concepts

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

Quantifiers
In propositional logic, quantifiers help to express the notion of 'there exists' or 'for all'. A common quantifier is the existential quantifier, denoted as \( \exists x \), which means 'there exists an x'. Another common quantifier is the universal quantifier, denoted as \( \forall x \), which means 'for all x'.

For example, \( \exists x P(x) \) means there is at least one x in the domain where the proposition \( P(x) \) is true. Conversely, \( \forall x P(x) \) means that the proposition \( P(x) \) is true for every x in the domain.

Quantifiers allow us to make precise and powerful statements about collections of objects in logic.
Negations
Negations in propositional logic are used to express the opposite of a given proposition. It is denoted by the symbol \( \eg \), which means 'not'.

If \( P(x) \) represents a proposition, then \( \eg P(x) \) means that the proposition is not true. For example, if \( P(1) \) says 'x is greater than 2', then \( \eg P(1) \) would say 'x is not greater than 2'.

Negations are essential for forming opposite or contradictory statements in logic. They help in constructing more complex propositions, making the logical relationships clearer and more comprehensive.
Disjunctions
Disjunctions are logical operations that combine two propositions with the word 'or'. It is expressed using the symbol \( \lor \). If we have two propositions \( P(x) \) and \( Q(x) \), their disjunction \( P(x) \lor Q(x) \) means that at least one of them is true.

For instance, \( \eg P(1) \lor \eg P(2) \lor \eg P(3) \lor \eg P(4) \lor \eg P(5) \) means that at least one of the propositions \( P(x) \) for x in the domain {1, 2, 3, 4, 5} is not true.

Disjunctions are beneficial for forming complex logical statements, where satisfying any one of multiple conditions is sufficient for the whole statement to be true.
Conjunctions
Conjunctions are logical operations that combine two propositions with the word 'and'. It is expressed using the symbol \( \land \). If we have two propositions \( P(x) \) and \( Q(x) \), their conjunction \( P(x) \land Q(x) \) means that both of them are true.

For instance, \( P(1) \land P(2) \land P(3) \land P(4) \land P(5) \) means that the propositions \( P(x) \) are true for all x in the domain {1, 2, 3, 4, 5}.

Conjunctions are significant in forming statements that require multiple conditions to be true simultaneously. They ensure that all combined propositions hold for the entire statement to be valid.

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Most popular questions from this chapter

Suppose that the domain of the propositional function \(P(x)\) consists of \(-5,-3,-1,1,3,\) and \(5 .\) Express these statements without using quantifiers, instead using only negations, disjunctions, and conjunctions. a) \(\exists x P(x)\) b) \(\quad \forall x P(x)\) c) \(\forall x((x \neq 1) \rightarrow P(x))\) d) \(\exists x((x \geq 0) \wedge P(x))\) e) \(\exists x(\neg P(x)) \wedge \forall x((x<0) \rightarrow P(x))\)

For each of these collections of premises, what relevant conclusion or conclusions can be drawn? Explain the rules of inference used to obtain each conclusion from the premises. a) 鈥淚f I take the day off, it either rains or snows.鈥� 鈥淚 took Tuesday off or I took Thursday off.鈥� 鈥淚t was sunny on Tuesday.鈥� 鈥淚t did not snow on Thursday.鈥� b) 鈥淚f I eat spicy foods, then I have strange dreams.鈥� 鈥淚 have strange dreams if there is thunder while I sleep.鈥� 鈥淚 did not have strange dreams.鈥� c) 鈥淚 am either clever or lucky.鈥� 鈥淚 am not lucky.鈥� 鈥淚f I am lucky, then I will win the lottery.鈥� d) 鈥淓very computer science major has a personal computer.鈥� 鈥淩alph does not have a personal computer.鈥� 鈥淎nn has a personal computer.鈥� e) 鈥淲hat is good for corporations is good for the United States.鈥� 鈥淲hat is good for the United States is good for you.鈥� 鈥淲hat is good for corporations is for you to buy lots of stuff.鈥� f ) 鈥淎ll rodents gnaw their food.鈥� 鈥淢ice are rodents.鈥� 鈥淩abbits do not gnaw their food.鈥� 鈥淏ats are not ro- dents.鈥�

Prove that if \(x\) is irrational, then 1\(/ x\) is irrational.

Use resolution to show that the compound proposition \((p \vee q) \wedge(\neg p \vee q) \wedge(p \vee \neg q) \wedge(\neg p \vee \neg q)\) is not satisfiable.

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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