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Chapter 1: Problem 25

Translate each of these statements into logical expressions using predicates, quantifiers, and logical connectives. a) No one is perfect. b) Not everyone is perfect. c) All your friends are perfect. d) At least one of your friends is perfect. e) Everyone is your friend and is perfect. f) Not everybody is your friend or someone is not perfect.

Short Answer

Expert verified
a) \( eg \exists x \; P(x) \), b) \( \exists x \; eg P(x) \), c) \( \forall x \; (F(x) \rightarrow P(x)) \), d) \( \exists x \; (F(x) \wedge P(x)) \), e) \( \forall x \; (F(x) \wedge P(x)) \), f) \( eg \forall x \; F(x) \vee \exists x \; eg P(x) \).

Step by step solution

01

- Define predicates

Let's define the predicates needed for the logical expressions. Let P(x) mean 'x is perfect' and F(x) mean 'x is your friend'.
02

- Translate statement a

Statement: No one is perfect.Logical translation: \[ eg \exists x \; P(x) \]This means there does not exist any x such that x is perfect.
03

- Translate statement b

Statement: Not everyone is perfect.Logical translation: \[ \exists x \; eg P(x) \]This means there exists at least one x such that x is not perfect.
04

- Translate statement c

Statement: All your friends are perfect.Logical translation: \[ \forall x \; (F(x) \rightarrow P(x)) \]This means for all x, if x is your friend, then x is perfect.
05

- Translate statement d

Statement: At least one of your friends is perfect.Logical translation: \[ \exists x \; (F(x) \wedge P(x)) \]This means there exists at least one x such that x is your friend and x is perfect.
06

- Translate statement e

Statement: Everyone is your friend and is perfect.Logical translation: \[ \forall x \; (F(x) \wedge P(x)) \]This means for all x, x is your friend and x is perfect.
07

- Translate statement f

Statement: Not everybody is your friend or someone is not perfect.Logical translation: \[ eg \forall x \; F(x) \vee \exists x \; eg P(x) \]This means it is not true that everyone is your friend or there exists at least one x such that x is not perfect.

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Key Concepts

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

predicates
In logic, predicates are functions that return truth values based on their input parameters. For example, consider the predicate P(x), where 'x is perfect'. This simply means that P(x) will be true if x is perfect and false otherwise. Predicates help us describe properties of objects within a particular domain. It鈥檚 like having a function that brings out specific characteristics of the objects involved.

Predicates can take various forms based on the scenario. In our exercise, we encountered two key predicates: P(x) and F(x).
  • P(x): Describes whether a given object x is perfect.
  • F(x): Describes whether a given object x is your friend.

By using predicates, we can translate natural language statements into more formal logical expressions that are easy to work with and analyze.
quantifiers
Quantifiers in logic specify the extent to which a predicate is true over a given domain. In our exercise, we used two primary quantifiers: the existential quantifier \(\exists\) and the universal quantifier \(\forall\).

  • \(\forall x P(x)\): This means 'for all x, P(x) is true'. It tells us that every object in our domain satisfies the predicate P(x).
  • \(\exists x P(x)\): This means 'there exists an x such that P(x) is true'. It indicates that at least one object in our domain satisfies the predicate P(x).

Let's connect this to the logical statements:
  • No one is perfect: \( eg \exists x P(x) \)
  • Not everyone is perfect: \(\exists x eg P(x)\)
  • All your friends are perfect: \(\forall x (F(x) \rightarrow P(x))\)

Quantifiers are critical in forming accurate and precise logical statements that match the meaning of natural language sentences.
logical connectives
Logical connectives are symbols used to connect predicates and form logical expressions. They help in building complex logical statements by specifying relationships between simpler ones. The most common logical connectives are:

  • \(eg\): Negation, which means 'not'. It inverts the truth value of a statement.
  • \(\wedge\): Conjunction, which means 'and'. It combines two statements, both of which must be true.
  • \(\vee\): Disjunction, which means 'or'. It combines two statements, at least one of which must be true.
  • \(\rightarrow\): Implication, which means 'if...then'. It denotes that if the first statement is true, then the second statement must also be true.

For instance:
  • All your friends are perfect: \(\forall x (F(x) \rightarrow P(x))\)
  • At least one of your friends is perfect: \(\exists x (F(x) \wedge P(x))\)

By using logical connectives, we can accurately represent complex logical relations in a precise and formal manner.

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Most popular questions from this chapter

Translate each of these nested quantifications into an English statement that expresses a mathematical fact. The domain in each case consists of all real numbers. a) \(\exists x \forall y(x+y=y)\) b) \(\forall x \forall y(((x \geq 0) \wedge(y<0)) \rightarrow(x-y>0))\) c) \(\exists x \exists y(((x \leq 0) \wedge(y \leq 0)) \wedge(x-y>0))\) d) \(\forall x \forall y((x \neq 0) \wedge(y \neq 0) \leftrightarrow(x y \neq 0))\)

Determine whether each of these arguments is valid. If an argument is correct, what rule of inference is being used? If it is not, what logical error occurs? a) If \(n\) is a real number such that \(n>1,\) then \(n^{2}>1\) Suppose that \(n^{2}>1 .\) Then \(n>1\) b) If \(n\) is a real number with \(n>3,\) then \(n^{2}>9\) . Suppose that \(n^{2} \leq 9 .\) Then \(n \leq 3\) . c) If \(n\) is a real number with \(n>2,\) then \(n^{2}>4\) . Suppose that \(n \leq 2 .\) Then \(n^{2} \leq 4 .\)

For each of these arguments, explain which rules of inference are used for each step. a) 鈥淒oug, a student in this class, knows how to write programs in JAVA. Everyone who knows how to write programs in JAVA can get a high-paying job. Therefore, someone in this class can get a high-paying job.鈥� b) 鈥淪omebody in this class enjoys whale watching. Every person who enjoys whale watching cares about ocean pollution. Therefore, there is a person in this class who cares about ocean pollution.鈥� c) 鈥淓ach of the 93 students in this class owns a personal computer. Everyone who owns a personal computer can use a word processing program. Therefore, Zeke, a student in this class, can use a word processing pro- gram.鈥� d) 鈥淓veryone in New Jersey lives within 50 miles of the ocean. Someone in New Jersey has never seen the ocean. Therefore, someone who lives within 50 miles of the ocean has never seen the ocean.鈥�

Prove that \(n^{2}+1 \geq 2^{n}\) when \(n\) is a positive integer with \(1 \leq n \leq 4 .\)

Express the negations of each of these statements so that all negation symbols immediately precede predicates. a) \(\forall x \exists y \forall z T(x, y, z)\) b) \(\forall x \exists y P(x, y) \vee \forall x \exists y Q(x, y)\) c) \(\forall x \exists y P(x, y) \wedge \exists z R(x, y, z) )\) d) \(\forall x \exists y(P(x, y) \rightarrow Q(x, y))\)
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