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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

Show that if \(a, b,\) and \(c\) are real numbers and \(a \neq 0\) , then there is a unique solution of the equation \(a x+b=c\) .

Express the negations of each of these statements so that all negation symbols immediately precede predicates. a) \(\forall x \exists y \forall z T(x, y, z)\) b) \(\forall x \exists y P(x, y) \vee \forall x \exists y Q(x, y)\) c) \(\forall x \exists y P(x, y) \wedge \exists z R(x, y, z) )\) d) \(\forall x \exists y(P(x, y) \rightarrow Q(x, y))\)

Express the negation of each of these statements in terms of quantifiers without using the negation symbol. a) \(\forall x(x>1)\) b) \(\forall x(x \leq 2)\) \(\begin{array}{ll}\text { c) } & \exists x(x \geq 4)\end{array}\) d) \(\exists x(x<0)\) e) \(\forall x((x<-1) \vee(x>2))\) f) \(\exists x((x<4) \vee(x>7))\)

A discrete mathematics class contains 1 mathematics major who is a freshman, 12 mathematics majors who are sophomores, 15 computer science majors who are sophomores, 2 mathematics majors who are juniors, 2 computer science majors who are juniors, and 1 computer science major who is a senior. Express each of these statements in terms of quantifiers and then determine its truth value. a) There is a student in the class who is a junior. b) Every student in the class is a computer science major. c) There is a student in the class who is neither a mathematics major nor a junior. d) Every student in the class is either a sophomore or a computer science major. e) There is a major such that there is a student in the class in every year of study with that major.

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.
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