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Chapter 1: Problem 3

What is the negation of each of these propositions? a) Linda is younger than Sanjay. b) Mei makes more money than Isabella. c) Moshe is taller than Monica. d) Abby is richer than Ricardo.

Short Answer

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a) Linda is not younger than Sanjay. b) Mei does not make more money than Isabella. c) Moshe is not taller than Monica. d) Abby is not richer than Ricardo.

Step by step solution

01

Understand Negation

Negation in logic means that you reverse the truth value of a proposition. If the original proposition is true, its negation is false, and vice versa.
02

Identify the Propositions

Identify and write down each proposition from the given exercise: a) Linda is younger than Sanjay. b) Mei makes more money than Isabella. c) Moshe is taller than Monica. d) Abby is richer than Ricardo.
03

Formulate the Negation of Each Proposition

Negate each proposition by changing the relationship: a) Linda is not younger than Sanjay. b) Mei does not make more money than Isabella. c) Moshe is not taller than Monica. d) Abby is not richer than Ricardo.

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Key Concepts

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

logical negation
Logical negation is a fundamental concept in propositional logic. It involves reversing the logical value of a proposition. If a proposition is true, its logical negation makes it false, and if it's false, the negation makes it true.
For example, consider the original proposition 'Linda is younger than Sanjay.' If this statement is true, the negation 'Linda is not younger than Sanjay' would be false. Similarly, if the original statement were false, the negation would be true.
Negation always flips the truth value. Here are some common terms associated with negation:
  • Not (卢): Used to represent the negation in logical expressions.
  • Opposite: Refers to the inverse truth value of the original proposition.
Understanding how to negate propositions is essential for solving logical problems and determining the truth of various statements.
propositional logic
Propositional logic deals with statements that can either be true or false. These statements are known as propositions. In propositional logic, propositions are connected using logical connectives like 'and,' 'or,' 'not,' etc.
Each proposition can have a truth value, and you can perform operations on these propositions to get new truth values. For instance:
  • In the proposition 'Mei makes more money than Isabella,' this is a single proposition that can be true or false.
  • When using logical connectives, you can form compound propositions, like 'Mei makes more money than Isabella and Moshe is taller than Monica.' This compound proposition is true only if both individual propositions are true.
Propositional logic helps in breaking down complex statements into simpler parts and analyzing their truth values systematically. It is widely used in mathematics, computer science, and philosophy.
truth value
The truth value of a proposition indicates whether the statement is true or false. In propositional logic, every proposition must have a definite truth value.
For example:
  • The proposition 'Abby is richer than Ricardo' has a truth value that can be either true or false based on the actual financial statuses of Abby and Ricardo.
  • When you negate a proposition like 'Abby is not richer than Ricardo,' the truth value is also flipped. If the original proposition is true, the negation is false, and vice versa.
Truth values are often represented using symbols:
  • True (T) or 1: Represents a true proposition
  • False (F) or 0: Represents a false proposition
Working with truth values is essential for logical reasoning, as it allows you to construct and evaluate arguments based on the truth or falsity of statements.

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Most popular questions from this chapter

Suppose the domain of the propositional function \(P(x, y)\) consists of pairs \(x\) and \(y,\) where \(x\) is \(1,2,\) or 3 and \(y\) is \(1,2,\) or \(3 .\) Write out these propositions using disjunctions and conjunctions. $$ \begin{array}{ll}{\text { a) } \forall x \forall y P(x, y)} & {\text { b) } \exists x \exists y P(x, y)} \\ {\text { c) } \exists x \forall y P(x, y)} & {\text { d) } \forall y \exists x P(x, y)}\end{array} $$

Use quantifiers and predicates with more than one variable to express these statements. a) Every computer science student needs a course in discrete mathematics. b) There is a student in this class who owns a personal computer. c) Every student in this class has taken at least one computer science course. d) There is a student in this class who has taken at least one course in computer science. e) Every student in this class has been in every building on campus. f ) There is a student in this class who has been in every room of at least one building on campus. g) Every student in this class has been in at least one room of every building on campus.

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.

Find a counterexample to the statement that every positive integer can be written as the sum of the squares of three integers.

Prove that these four statements about the integer \(n\) are equivalent: \((i) n^{2}\) is odd, \((i i) 1-n\) is even, \((i i i) n^{3}\) is odd (iv) \(n^{2}+1\) is even.
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