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Chapter 1: Problem 38

Express the negations of these propositions using quantifiers, and in English. a) Every student in this class likes mathematics. b) There is a student in this class who has never seen a computer. c) There is a student in this class who has taken every mathematics course offered at this school. d) There is a student in this class who has been in at least one room of every building on campus.

Short Answer

Expert verified
a) There is a student in this class who does not like mathematics. b) Every student in this class has seen a computer. c) Every student in this class has not taken at least one mathematics course offered at this school. d) Every student in this class has not been in at least one room of every building on campus.

Step by step solution

01

Understand the quantifiers and negations

The two main quantifiers used in logic are 'for all' (\( \forall \)) and 'there exists' (\( \thereexists \)). Negating these quantifiers involves converting 'for all' to 'there exists not' (eq \forall x, P(x)) and 'there exists' to 'for all not' (\( \forall x, eg P(x) \)).
02

Negate the first proposition

'Every student in this class likes mathematics' can be written as \( \forall x (S(x) \rightarrow M(x)) \). The negation is \( eg \forall x (S(x) \rightarrow M(x)) \), which is equivalent to \( \thereexists x eg (S(x) \rightarrow M(x)) \). Since \( S(x) \rightarrow M(x) \) is equivalent to \( eg S(x) \text{ or } M(x) \), the negation becomes \( \thereexists x (S(x) \text{ and } eg M(x)) \). In English, this means 'There is a student in this class who does not like mathematics.'
03

Negate the second proposition

'There is a student in this class who has never seen a computer' can be written as \( \thereexists x (S(x) \text{ and } eg C(x)) \). The negation is \( eg \thereexists x (S(x) \text{ and } eg C(x)) \), which is equivalent to \( \forall x eg (S(x) \text{ and } eg C(x)) \). Using De Morgan's laws, this is equivalent to \( \forall x ( eg S(x) \text{ or } C(x)) \). In English, this means 'Every student in this class has seen a computer.'
04

Negate the third proposition

'There is a student in this class who has taken every mathematics course offered at this school' can be written as \( \thereexists x (S(x) \text{ and } \forall y (M(y) \rightarrow T(x,y))) \). The negation is \( eg \thereexists x (S(x) \text{ and } \forall y (M(y) \rightarrow T(x,y))) \), which is equivalent to \( \forall x eg (S(x) \text{ and } \forall y (M(y) \rightarrow T(x,y))) \). Using De Morgan's laws, this becomes \( \forall x ( eg S(x) \text{ or } \thereexists y (M(y) \text{ and } eg T(x,y))) \). In English, this means 'Every student in this class has not taken at least one mathematics course offered at this school.'
05

Negate the fourth proposition

'There is a student in this class who has been in at least one room of every building on campus' can be written as \( \thereexists x (S(x) \text{ and } \forall y (B(y) \rightarrow R(x,y))) \). The negation is \( eg \thereexists x (S(x) \text{ and } \forall y (B(y) \rightarrow R(x,y))) \), which is equivalent to \( \forall x eg (S(x) \text{ and } \forall y (B(y) \rightarrow R(x,y))) \). Using De Morgan's laws, this becomes \( \forall x ( eg S(x) \text{ or } \thereexists y (B(y) \text{ and } eg R(x,y))) \). In English, this means 'Every student in this class has not been in at least one room of every building on campus.'

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Key Concepts

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

Logical Quantifiers
Logical quantifiers are symbols used in logical expressions to denote the extent to which a predicate applies to a subject.
The two most common quantifiers are:\( \forall \) (for all) and \( \exists \) (there exists).
  • \( \forall \) indicates that the predicate applies to all elements in a particular set. For example, 'Every student in this class likes mathematics' can be represented as \( \forall x \) (S(x) implies M(x)), where S(x) means 'x is a student in this class' and M(x) means 'x likes mathematics.'

  • \( \exists \) signifies that there is at least one element in the set for which the predicate is true. For example, 'There is a student in this class who has never seen a computer' can be expressed as \( \exists x \) (S(x) and not C(x)), where C(x) means 'x has seen a computer.'
Understanding how to use these quantifiers is key to forming and negating logical propositions.
De Morgan's Laws
De Morgan's laws are essential rules in logic that help transform the negation of a statement involving logical quantifiers. These laws provide a systematic way to negate logical expressions by distributing the negation through the expression.
  • The first law states: The negation of a conjunction (AND) is equivalent to the disjunction (OR) of the negations. In symbolic form: \[eg (P \land Q) \equiv (eg P) \lor (eg Q)\]
  • The second law specifies: The negation of a disjunction (OR) is equivalent to the conjunction (AND) of the negations. In symbolic form: \[eg (P \lor Q) \equiv (eg P) \land (eg Q)\]
These laws are crucial for understanding how to negate complex logical statements.
For example, when negating 'There is a student in this class who has taken every mathematics course,' we use De Morgan's laws to transform the negation correctly. Applying these laws helps us state the negation as 'Every student in this class has not taken at least one mathematics course.'
Predicate Logic
Predicate logic, also known as first-order logic, is an extension of propositional logic.
It includes quantifiers and predicates that allow for a more detailed analysis of statements.
Predicate logic is used to express propositions involving objects and their properties. It connects objects in a domain with predicates to form statements.
  • For instance, \(P(x)\) could represent 'x is a prime number,' while \(Q(x)\) might mean 'x is even.'
    Combining these with logical quantifiers enriches the expressiveness of our statements.
  • In predicate logic, a statement like 'Some numbers are prime' can be written as \(\exists x P(x)\), where the predicate \(P(x)\) expresses that x is a prime number.
By incorporating structured symbols and logical operators, predicate logic improves our ability to handle and understand complex propositions. This approach is pivotal when negating expressions or dealing with conditions involving multiple objects or scenarios.

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Most popular questions from this chapter

Let \(C(x)\) be the statement " \(x\) has a cat," let \(D(x)\) be the statement " \(x\) has a dog," and let \(F(x)\) be the statement "x has a ferret." Express each of these statements in terms of \(C(x), D(x), F(x),\) quantifiers, and logical connectives. Let the domain consist of all students in your class. a) A student in your class has a cat, a dog, and a ferret. b) All students in your class have a cat, a dog, or a ferret. c) Some student in your class has a cat and a ferret, but not a dog. d) No student in your class has a cat, a dog, and a ferret. e) For each of the three animals, cats, dogs, and ferrets, there is a student in your class who has this animal as a pet.

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 duck,鈥� 鈥渪 is one of my poultry,鈥� 鈥渪 is an officer,鈥� and 鈥渪 is willing to waltz,鈥� respectively. Express each of these statements using quantifiers; logical connectives; and P(x), Q(x), R(x), and S(x). a) No ducks are willing to waltz. b) No officers ever decline to waltz. c) All my poultry are ducks. d) My poultry are not officers. e) Does (d) follow from (a), (b), and (c)? If not, is there a correct conclusion?

Exercises \(40-44\) deal with the translation between system specification and logical expressions involving quantifiers. Express each of these system specifications using predicates, quantifiers, and logical connectives. a) At least one mail message, among the nonempty set of messages, can be saved if there is a disk with more than 10 kilobytes of free space. b) Whenever there is an active alert, all queued messages are transmitted. c) The diagnostic monitor tracks the status of all systems except the main console. d) Each participant on the conference call whom the host of the call did not put on a special list was billed.

Use resolution to show that the hypotheses 鈥淚t is not raining or Yvette has her umbrella,鈥� 鈥淵vette does not have her umbrella or she does not get wet,鈥� and 鈥淚t is raining or Yvette does not get wet鈥� imply that 鈥淵vette does not get wet.鈥�

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