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Chapter 1: Problem 13

Determine the truth value of each of these statements if the domain consists of all integers. $$ \begin{array}{ll}{\text { a) } \forall n(n+1>n)} & {\text { b) } \exists n(2 n=3 n)} \\ {\text { c) } \exists n(n=-n)} & {\text { d) } \forall n(3 n \leq 4 n)}\end{array} $$

Short Answer

Expert verified
Statements a) and d) are true for all integers. Statements b) and c) are true for integer n=0.

Step by step solution

01

Understanding Universal Quantifiers

Statement a) \( \forall n \big( n + 1 > n \big) \) needs to be true for all integers. Consider if adding 1 to any integer results in a number greater than the original integer.
02

Solution for Part a

Since for any integer \( n \), \( n + 1 \) will always be greater than \( n \), the statement \( \forall n \big( n + 1 > n \big) \) is true.
03

Understanding Existential Quantifiers

Statement b) \( \big( \therefore exists n \big) (2n = 3n) \) needs to check if there exists any integer \( n \) that satisfies the equation \( 2n = 3n \).
04

Solution for Part b

Rewriting \( 2n = 3n \) gives \( n = 0 \). Since \( n = 0 \) is an integer, the statement \( \therefore exists n \big( 2n = 3n \) is true.
05

Evaluating Zero Identity

Statement c) \( \big( \therefore exists n \big) (n = -n) \) checks if there exists an integer \( n \) such that the integer equals its negative.
06

Solution for Part c

The equation \( n = -n \) simplifies to \( 2n = 0 \) or \( n = 0 \). Therefore, \( n = 0 \) exists and is an integer.
07

Statement Combining Constants

Statement d) \( \forall n \big( 3n eq 4n) \) verifies if for all integers \( n \), is \( 3n <= 4n \).
08

Solution for Part d

Since for any integer \( n \), multiplying it by 3 will always be less than or equal to multiplying it by 4, the statement \( \forall n \big( 3n <= 4n \) is true.

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Key Concepts

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

Universal Quantifier
The universal quantifier is a crucial concept in discrete mathematics. It is denoted by the symbol \( \forall \), which means 'for all'. When using the universal quantifier, a statement must be true for every element within a given domain.

For example, in the problem statement \( \forall n (n + 1 > n) \), we are asked to determine if adding 1 to any integer always results in a greater number compared to the original. Since this holds true for all integers, the statement is true.

Thus, the universal quantifier ensures that a property or condition applies universally to all elements in the domain.
Existential Quantifier
The existential quantifier focuses on the existence of at least one element in the domain that satisfies a given condition. It is represented by the symbol \( \therefore \), which translates to 'there exists'.

For instance, in the statement \( \therefore n (2n = 3n) \), we are checking if there is at least one integer \( n \) that makes the equation true. By simplifying, we find that \( n = 0 \) satisfies this equation. Therefore, the statement is true.

The existential quantifier helps identify specific elements within a domain that fulfill a particular requirement.
Truth Value Evaluation
Evaluating the truth value of statements involves verifying whether the conditions expressed by quantifiers hold true within a specified domain. We often break down the process into systematic steps:
  • Identify the type of quantifier (universal or existential)
  • Translate the mathematical statement and simplify if needed
  • Confirm if the statement is true for all or some elements in the domain

For example, evaluating \( \therefore n (n = -n) \) means checking if there is any integer \( n \) equal to its negative. Simplifying gives \( 2n = 0 \) or \( n = 0 \). Thus, the statement is true as it holds for \( n = 0 \).

This systematic approach helps in clear and effective truth value evaluation.
Integer Domain
The integer domain refers to the set of all integers, both positive and negative, including zero. In evaluating statements, we need to consider this entire set. For example, in the statement \( \forall n (3n <= 4n) \), we verify if multiplying any integer by 3 is always less than or equal to multiplying it by 4. This holds true for all integers \( n \).

working with the integer domain requires:
  • Understanding the properties of integers
  • Considering the inclusiveness of both positive and negative values, as well as zero

Recognizing the scope of the integer domain is key to accurately determining the truth value of mathematical statements.

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Most popular questions from this chapter

Prove that there are no solutions in integers \(x\) and \(y\) to the equation \(2 x^{2}+5 y^{2}=14 .\)

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. $$ \begin{array}{ll}{\text { a) } \exists x P(x, 3)} & {\text { b) } \forall y P(1, y)} \\ {\text { c) } \exists y \neg P(2, y)} & {\text { d) } \forall x \neg P(x, 2)}\end{array} $$

Express each of these statements using quantifiers. Then form the negation of the statement, so that no negation is to the left of a quantifier. Next, express the negation in simple English. (Do not simply use the phrase "It is not the case that.") a) Some old dogs can learn new tricks. b) No rabbit knows calculus. c) Every bird can fly. d) There is no dog that can talk. e) There is no one in this class who knows French and Russian.

Prove or disprove that you can use dominoes to tile the standard checkerboard with two adjacent corners removed (that is, corners that are not opposite).

Express each of these statements using quantifiers. Then form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English. (Do not simply use the phrase "It is not the case that.") a) No one has lost more than one thousand dollars playing the lottery. b) There is a student in this class who has chatted with exactly one other student. c) No student in this class has sent e-mail to exactly two other students in this class. d) Some student has solved every exercise in this book. e) No student has solved at least one exercise in every section of this book.
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