/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 7 Let \(R(m, n)\) be the predicate... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 7

Let \(R(m, n)\) be the predicate "lf \(m\) is a factor of \(n^{2}\) then \(m\) is a factor of \(n, "\) with domain for both \(m\) and \(n\) being the set \(\mathbf{Z}\) of integers. a. Explain why \(R(m, n)\) is false if \(m=25\) and \(n=10\). b. Give values different from those in part (a) for which \(R(m, n)\) is false. c. Explain why \(R(m, n)\) is true if \(m=5\) and \(n=10\).

Short Answer

Expert verified
a. R(m, n) is false if m=25 and n=10 because 25 is a factor of \(n^2 = 100\), but 25 is not a factor of n=10. b. R(m, n) is also false for m=49 and n=14, as 49 is a factor of \(n^2 = 196\), but 49 is not a factor of n=14. c. R(m, n) is true if m=5 and n=10 because 5 is a factor of \(n^2 = 100\) and 5 is a factor of n=10.

Step by step solution

01

Part a - Test R(25, 10)

In this part, we need to find if R(m, n) is false if m=25 and n=10. First, we need to check if m is a factor of \(n^{2}\). n^2: \(10^2 = 100\) Now, since m=25, let's determine if 25 is a factor of 100. 100 梅 25 = 4, which is an integer. So, 25 is a factor of 100. Now, let's verify if 25 is a factor of 10 (n). 10 梅 25 = 0.4, which is not an integer. Hence, 25 is not a factor of 10. Since "m is a factor of n^2" holds true, but "m is a factor of n" does not hold true, R(25, 10) is false.
02

Part b - Find other false values for R(m, n)

To find other false values for R(m, n), we can start by finding a value of m such that it is a factor of n^2 but not a factor of n. One example can be: m=49 and n=14 First, we need to check if m is a factor of \(n^{2}\). n^2: \(14^2 = 196\) Now, since m=49, let's determine if 49 is a factor of 196. 196 梅 49 = 4, which is an integer. So, 49 is a factor of 196. Now, let's verify if 49 is a factor of 14 (n). 14 梅 49 鈮� 0.29, which is not an integer. Hence, 49 is not a factor of 14. Since "m is a factor of n^2" holds true, but "m is a factor of n" does not hold true, R(49, 14) is false.
03

Part c - Test R(5, 10)

In this part, we need to find if R(m, n) is true if m=5 and n=10. First, we need to check if m is a factor of \(n^{2}\). n^2: \(10^2 = 100\) Now, since m=5, let's determine if 5 is a factor of 100. 100 梅 5 = 20, which is an integer. So, 5 is a factor of 100. Now, let's verify if 5 is a factor of 10 (n). 10 梅 5 = 2, which is an integer. Hence, 5 is a factor of 10. Since "m is a factor of n^2" holds true, and "m is a factor of n" also holds true, R(5, 10) is true.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Number Theory
Number Theory plays a pivotal role in understanding the exercise involving factors of numbers. In this case, we examine if a number, say \( m \), is a factor of another number \( n^2 \), and if it subsequently needs to be a factor of \( n \). Number Theory helps us explore properties such as divisibility, factors, and multiples.
When a number \( a \) divides another number \( b \) without any remainder, we say \( a \) is a factor of \( b \). For example, \( 5 \) is a factor of \( 10 \) because when you divide \( 10 \) by \( 5 \), the result is \( 2 \), which is a whole number. However, factors have more to them when observed in the square of a number.
  • Take \( n^2 \): When a number is squared, its factors become more diversified. Some numbers can be factors of \( n^2 \) but not necessarily of \( n \) itself.
  • This distinction is where Number Theory helps us test and find counterexamples, as seen with \( m = 25 \) and \( n = 10 \).
Number Theory equips us with tools to strategically check such mathematical relationships.
Predicate Logic
Predicate Logic is a framework used to form expressions that can be true or false depending on variable replacement. It allows for complex structures in logic that aren't suitable for basic propositional logic.
Predicate logic consists of predicates and quantifiers and can express statements involving one or more variables. In this exercise, we come across a statement \( R(m, n) \), where \( m \) and \( n \) are variables within a domain of integers.
  • The predicate \( R(m, n) \) is either true or false based on whether the statement "\( m \) is a factor of \( n^2 \) then \( m \) is a factor of \( n \)" holds for specific integers \( m \) and \( n \).
  • Predicate logic allows the formulation of such conditions so that one can systematically verify the truth of logical expressions using various examples and counterexamples.
This logical exploration aids in understanding how precise conditions can affect the validity of the expression for different numerical values.
Factors in Mathematics
Factors in Mathematics are central to the problem we are analyzing. A factor is a number that divides another number completely without leaving a remainder. Understanding and identifying factors is essential for many mathematical operations and concepts.
In this exercise, we explore how determining factors of numbers \( n^2 \) and \( n \) helps us check the validity of \( R(m, n) \).
  • For example, the number \( 25 \) is a factor of \( 100 \) because \( 25 \times 4 = 100 \). However, \( 25 \) is not a factor of \( 10 \), demonstrating how the square \( n^2 \) can have additional factors compared to \( n \).
  • This concept assists in understanding the different outcomes of the function \( R(m, n) \) depending on whether \( m \) truly divides \( n \) as it does \( n^2 \).
Understanding such nuances enriches students鈥� insights into factor relationships across different cases in mathematics.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Let the domain of \(x\) be the set \(D\) of objects discussed in mathematics courses, and let Real \((x)\) be " \(x\) is a real number," \(\operatorname{Pos}(x)\) be \({ }^{*} x\) is a positive real number," \(\operatorname{Neg}(x)\) be " \(x\) is a negative real number," and \(\operatorname{Int}(x)\) be " \(x\) is an integer." a. \(\operatorname{Pos}(0)\) b. \(\forall x\), Real \((x) \wedge \operatorname{Neg}(x) \rightarrow \operatorname{Pos}(-x)\). c. \(\forall x, \operatorname{Int}(x) \rightarrow \operatorname{Real}(x)\). d. \(\exists x\) such that \(\operatorname{Real}(x) \wedge \sim \operatorname{lnt}(x)\).

Give an example to show that a universal conditional statement is not logically equivalent to its inverse. Earning a grade of \(\mathrm{C}\) - in this course is a sufficient condition for it to count toward graduation.

Informal language is actually more complex than formal language. That is what makes the job of a systems analyst so challenging. A systems analyst works as an intermediary between a client who uses informal language and a programmer who needs precise specifications in order to produce code. For instance, the sentence "There are no orders from store \(A\) for item \(B\) " contains the words there are. Is the statement existential? Write an informal negation for the statement, and then write the statement formally using quantifiers and variables.

In exercises \(28-33\), reorder the premises in each of the arguments to show that the conclusion follows as a valid consequence from the premises. It may be helpful to rewrite the statements in ifthen form and replace some statements by their contrapositives. Exercises 28-30 refer to the kinds of Tarski worlds discussed in Example 2.1.12 and 2.3.1. Exercises 31 and 32 are adapted from Symbolic Logic by Lewis Carroll.* 1\. I trust every animal that belongs to me. 2\. Dogs gnaw bones. 3\. I admit no animals into my study unless they will beg when told to do so. 4\. All the animals in the yard are mine. 5\. I admit every animal that I trust into my study. 6\. The only animals that are really willing to beg when told to do so are dogs. \(\therefore\) All the animals in the yard gnaw bones.

In each of 14-19, (a) rewrite the statement in English without using the symbol \(\forall\) or \(\exists\) but expressing your answer as simply as possible, and (b) write a negation for the statement. \(\forall\) colors \(C, \exists\) an animal \(A\) such that \(A\) is colored \(C\).
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.