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Chapter 1: Problem 23

Translate in two ways each of these statements into logical expressions using predicates, quantifiers, and logical connectives. First, let the domain consist of the students in your class and second, let it consist of all people. a) Someone in your class can speak Hindi. b) Everyone in your class is friendly. c) There is a person in your class who was not born in California. d) A student in your class has been in a movie. e) No student in your class has taken a course in logic programming.

Short Answer

Expert verified
a) \(\exists x \, ( H(x) )\), \(\exists x \, ( S(x) \land H(x) )\); b) \(\forall x \, ( F(x) )\), \(\forall x \, ( S(x) \rightarrow F(x) )\); c) \(\exists x \, ( eg C(x) )\), \(\exists x \, ( S(x) \land eg C(x) )\); d) \(\exists x \, ( M(x) )\), \(\exists x \, ( S(x) \land M(x) )\); e) \(\forall x \, ( eg L(x) )\), \(\forall x \, ( S(x) \rightarrow eg L(x) )\).

Step by step solution

01

Define Symbols and Predicates

Define the necessary symbols and predicates for representing the statements.For all students in the class:- Let the domain be students in the class.- Let the predicate be: - H(x): x can speak Hindi - F(x): x is friendly - C(x): x was born in California - M(x): x has been in a movie - L(x): x has taken a course in logic programmingFor all people:- Let the domain be all people.- Let the predicate be defined for the same properties.
02

Part (a) - Someone in your class can speak Hindi

1. Domain: Students in your class - Existential quantifier - Logical expression: \[ \exists x \, ( H(x) ) \] 2. Domain: All people - Existential quantifier with condition that x is a student in your class - Logical expression: \[ \exists x \, ( S(x) \land H(x) ) \]
03

Part (b) - Everyone in your class is friendly

1. Domain: Students in your class - Universal quantifier - Logical expression: \[ \forall x \, ( F(x) ) \] 2. Domain: All people - Universal quantifier with condition that x is a student in your class - Logical expression: \[ \forall x \, ( S(x) \rightarrow F(x) ) \]
04

Part (c) - There is a person in your class who was not born in California

1. Domain: Students in your class - Existential quantifier and negation - Logical expression: \[ \exists x \, ( eg C(x) ) \] 2. Domain: All people - Existential quantifier with negation and condition that x is a student in your class - Logical expression: \[ \exists x \, ( S(x) \land eg C(x) ) \]
05

Part (d) - A student in your class has been in a movie

1. Domain: Students in your class - Existential quantifier - Logical expression: \[ \exists x \, ( M(x) ) \] 2. Domain: All people - Existential quantifier with condition that x is a student in your class - Logical expression: \[ \exists x \, ( S(x) \land M(x) ) \]
06

Part (e) - No student in your class has taken a course in logic programming

1. Domain: Students in your class - Universal quantifier and negation - Logical expression: \[ \forall x \, ( eg L(x) ) \] 2. Domain: All people - Universal quantifier with negation and condition that x is a student in your class - Logical expression: \[ \forall x \, ( S(x) \rightarrow eg L(x) ) \]

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Key Concepts

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

logical expressions
Logical expressions are used to articulate specific statements and arguments using logical symbols. These expressions help in representing complex ideas in a simplified, mathematical form. They comprise symbols like \( \forall\ \, \ \exists\ \), connectives, and predicates. Understanding logical expressions helps in formal reasoning and in transforming statements into a formal language.
predicates
Predicates are fundamental components in logical expressions. They are functions that denote properties or relationships between objects. For example, in the predicate \( F(x) \), \(F\) represents a property (e.g., 'is friendly'), and \(x\) is the subject. Predicates help break down complex statements into manageable parts, making it easier to apply quantifiers and logical connectives.
quantifiers
Quantifiers specify the quantity of subjects to which a predicate applies. They are essential in forming logical expressions. There are two main types of quantifiers:
  • Universal Quantifier ( \( \forall \) ): Indicates that a predicate applies to all elements in a domain.
  • Existential Quantifier ( \( \exists \) ): Indicates that there exists at least one element in the domain for which the predicate is true.
Understanding quantifiers is crucial for correctly interpreting and formulating logical statements.
logical connectives
Logical connectives are symbols that link statements together to form a logical expression. The most common connectives are:
  • Conjunction ( \( \land \) ): Represents 'and'. Both statements must be true.
  • Disjunction ( \( \lor \) ): Represents 'or'. At least one statement must be true.
  • Negation ( \( \eg \) ): Represents 'not'. Inverts the truth value of a statement.
  • Implication ( \(( \rightarrow )\): Represents 'if ... then'. If the first statement is true, then the second must be true.
These connectives are essential tools for constructing more complex logical expressions and performing logical operations.
universal quantifier
The universal quantifier, denoted by \( \forall \), is used to indicate that a predicate holds for all elements within a particular domain. For example, the statement 'Everyone in your class is friendly' can be expressed using a universal quantifier: \[ \forall x ( F(x) ) \] where \(x\) represents any student in the class and \(F(x)\) indicates 'is friendly'. This quantifier is crucial for expressing comprehensive statements about an entire group.
existential quantifier
The existential quantifier, denoted by \( \exists \), is used to assert that there is at least one element in the domain for which the predicate is true. For instance, 'Someone in your class can speak Hindi' translates to: \[ \exists x ( H(x) ) \] where \(x\) is any student in the class and \(H(x)\) means 'can speak Hindi'. The existential quantifier helps in expressing statements about the existence of specific instances within a broader group.

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Most popular questions from this chapter

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