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Chapter 1: Problem 9

Let L(x, y) be the statement 鈥渪 loves y,鈥� where the domain for both x and y consists of all people in the world. Use quantifiers to express each of these statements. a) Everybody loves Jerry. b) Everybody loves somebody. c) There is somebody whom everybody loves. d) Nobody loves everybody. e) There is somebody whom Lydia does not love. f ) There is somebody whom no one loves. g) There is exactly one person whom everybody loves. h) There are exactly two people whom Lynn loves. i) Everyone loves himself or herself. j) There is someone who loves no one besides himself or herself.

Short Answer

Expert verified
a) \(\forall x (L(x, j))\), b) \(\forall x \exists y (L(x, y))\), c) \(\exists y \forall x (L(x, y))\), d) \(eg \exists x \forall y (L(y, x))\), e) \(\exists y \eg (L(l, y))\), f) \(\exists y \forall x \eg (L(x, y))\), g) \(\exists y ( \forall x (L(x, y)) \land \forall z ( \forall x (L(x, z)) \rightarrow z = y ) )\), h) \(\exists y_1 \exists y_2 (L(l, y_1) \land L(l, y_2) \land y_1 \eq y_2 \land \forall y ( L(l, y) \rightarrow (y = y_1 \or y = y_2) ) )\), i) \(\forall x (L(x, x))\), j) \{\exists x ( \forall y (L(x, y) \leftrightarrow x = y) )}\.

Step by step solution

01

Statement Analysis

Interpret each statement and identify the appropriate quantifiers and logical connectors needed.
02

Quantifiers and Logical Connectors

Understand that \(\forall\) represents 'for all' and \(\exists\) represents 'there exists.' Logical connectors such as \(\rightarrow\) for 'implies' and \(eg\) for 'not' will be needed.
03

Express Statement a

a) For everybody (\( x \)) in the world, they love Jerry (\( j \)): \( \forall x (L(x, j)) \)
04

Express Statement b

b) For everybody (\( x \)) in the world, there exists somebody (\( y \)) whom they love: \( \forall x \exists y (L(x, y)) \)
05

Express Statement c

c) There exists somebody (\( y \)) whom everybody (\( x \)) in the world loves: \( \exists y \forall x (L(x, y)) \)
06

Express Statement d

d) There is no person (\( x \)) who everybody (\( y \)) loves: \( \eg \exists x \forall y (L(y, x)) \)
07

Express Statement e

e) There exists somebody (\( y \)) whom Lydia (\( l \)) does not love: \( \exists y \eg (L(l, y)) \)
08

Express Statement f

f) There exists somebody (\( y \)) whom nobody (\( x \)) loves: \( \exists y \forall x \eg (L(x, y)) \)
09

Express Statement g

g) There is exactly one person (\( y \)) whom everybody (\( x \)) loves: \( \exists y [ \forall x (L(x, y)) \land \forall z ( \forall x (L(x, z)) \rightarrow z = y ) ] \)
10

Express Statement h

h) There are exactly two people (\( y_1 \) and \( y_2 \)) whom Lynn (\( l \)) loves: \( \exists y_1 \exists y_2 (L(l, y_1) \land L(l, y_2) \land y_1 \eq y_2 \land \forall y ( L(l, y) \rightarrow (y = y_1 \lor y = y_2) )) \)
11

Express Statement i

i) Everybody (\( x \)) loves themselves: \( \forall x (L(x, x)) \)
12

Express Statement j

j) There is someone (\( x \)) who loves no one besides themselves: \( \exists x ( \forall y (L(x, y) \leftrightarrow x = y) ) \)

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Key Concepts

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

Mathematical Logic
Mathematical logic is the formal logical system used to describe mathematical concepts and reasoning. It provides a framework for expressing mathematical propositions and validating arguments through formal proofs. This area of logic uses symbolic representations to convey meaning and relationships.
For example, symbols like \( \forall \) and \( \exists \) help to represent 'for all' and 'there exists,' which are essential in expressing wide-ranging mathematical statements.
Mathematical logic is divided into several branches, such as:
  • **Propositional Logic**: Deals with propositions which can be true or false.
  • **Predicate Logic**: Goes beyond individual propositions and involves predicates and quantifiers.
  • **Set Theory**: Studies collections of objects and their relationships.
Understanding mathematical logic forms the foundation for advancing in fields like computer science, mathematics, and philosophy.
The symbols and operations from mathematical logic allow for precise communication of ideas that might be cumbersome in natural language.
Predicate Logic
Predicate logic, also known as first-order logic, dives deeper than the basic true-or-false nature of propositional logic. It involves quantifiers, predicates, and subjects. A predicate expresses a property that a subject can have, and it often looks like a function applied to an object.
For example, in the predicate \(L(x, y) \), 'L' might represent 'loves,' 'x' is the subject who loves, and 'y' is the object being loved. Predicates can be combined with quantifiers to build more complex logical statements. Two primary quantifiers are:
  • **Universal Quantifier \( \forall \)**: Indicates that the statements hold for all elements in the domain. Example: \( \forall x L(x, j) \) means 'everybody loves Jerry.'
  • **Existential Quantifier \( \exists \)**: Suggests that the statement holds for at least one element in the domain. Example: \( \exists x L(x, y) \) means 'there exists someone who loves y.'
Combining these elements allows us to articulate complex ideas succinctly and precisely.
Logical Quantifiers
Logical quantifiers are the backbone of expressing ideas in predicate logic and mathematical logic. They help in building statements that can either generalize or specify assertions about objects.
The two primary quantifiers are:
  • **Universal Quantifier ( \( \forall \) )**: Reads as 'for all,' this quantifier asserts that a proposition is true for every element in the domain set. Example: \( \forall x (L(x, j)) \) means 'everyone loves Jerry.'
  • **Existential Quantifier ( \( \exists \) )**: Reads as 'there exists,' it asserts that there is at least one element in the domain that fulfills the proposition. Example: \( \exists x (L(x, y)) \) means 'there's someone who loves y.'
Quantifiers can significantly alter the proposition's meaning when combined with logical operators such as 'not' ( \( eg \) ) or 'implies' ( \( \rightarrow \) ). For instance,
  • **Nobody loves everybody**: \( eg \exists x \forall y (L(y, x)) \) means it's not true that there's someone whom everyone loves.
  • **There is somebody whom no one loves**: \( \exists x \forall y ( \eg L(y, x) ) \).
Mastering logical quantifiers allows one to express various complex ideas in a structured and understandable manner.

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Most popular questions from this chapter

For each of these sets 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 play hockey, then I am sore the next day.鈥� 鈥淚 use the whirlpool if I am sore.鈥� 鈥淚 did not use the whirlpool.鈥� b) 鈥淚f I work, it is either sunny or partly sunny.鈥� 鈥淚 worked last Monday or I worked last Friday.鈥� 鈥淚t was not sunny on Tuesday.鈥� 鈥淚t was not partly sunny on Friday.鈥� c) 鈥淎ll insects have six legs.鈥� 鈥淒ragonflies are insects.鈥� 鈥淪piders do not have six legs.鈥� 鈥淪piders eat dragon-flies.鈥� d) 鈥淓very student has an Internet account.鈥� 鈥淗omer does not have an Internet account.鈥� 鈥淢aggie has an Internet account.鈥� e) 鈥淎ll foods that are healthy to eat do not taste good.鈥� 鈥淭ofu is healthy to eat.鈥� 鈥淵ou only eat what tastes good.鈥� 鈥淵ou do not eat tofu.鈥� 鈥淐heeseburgers are not healthy to eat.鈥� f ) 鈥淚 am either dreaming or hallucinating.鈥� 鈥淚 am not dreaming.鈥� 鈥淚f I am hallucinating, I see elephants running down the road.鈥�

Prove that \(m^{2}=n^{2}\) if and only if \(m=n\) or \(m=-n\)

Suppose that \(a\) and \(b\) are odd integers with \(a \neq b .\) Show there is a unique integer \(c\) such that \(|a-c|=|b-c|\)

Use quantifiers and predicates with more than one variable to express these statements. a) There is a student in this class who can speak Hindi. b) Every student in this class plays some sport. c) Some student in this class has visited Alaska but has not visited Hawaii. d) All students in this class have learned at least one programming language. e) There is a student in this class who has taken every course offered by one of the departments in this school. f ) Some student in this class grew up in the same town as exactly one other student in this class. g) Every student in this class has chatted with at least one other student in at least one chat group.

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 \(n^{2} \leq 4 .\)
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