/*! 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 8 Suppose that \(p, q\), and \(r\)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 8

Suppose that \(p, q\), and \(r\) are natural numbers such that \(p \cdot q

Short Answer

Expert verified
Given that \(p\) is a positive natural number, dividing the inequality \(p \cdot q < p \cdot r\) by \(p\), we directly obtain \(q<r\).

Step by step solution

01

Setup the Given Inequality

We start by looking at the inequality given by the exercise, which says that for natural numbers, if we have the inequality replacing it with the terms from the exercise.
02

Manipulate the Inequality

Since we know that the numbers are natural, we can assume that \(p\) must be positive. Thus we can divide both sides of the inequality by \(p\) without changing the direction of the inequality.
03

Simplify to Reveal the Relationship

After dividing both sides by \(p\), the \(p\)'s cancel out on both sides, leaving us with the inequality \(q < r\).
04

Conclude the Finding

Since the inequality simplifies directly to \(q<r\) without making any assumptions apart from that \(p\) is a natural number, and therefore positive, we have shown that \(q<r\).

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.

Natural Numbers
Natural numbers are the set of positive integers that we use in everyday counting. They start from 1 and go upwards indefinitely, so the set includes numbers like 1, 2, 3, etc. In mathematical notation, we typically represent the set of natural numbers with the symbol \( \mathbb{N} \).

It's essential to know that natural numbers do not include zero or negative numbers; they are strictly positive. This characteristic has implications in many areas of mathematics, including inequality proofs. Since natural numbers are always positive, any product of natural numbers is also positive. This plays a crucial role in simplifying and understanding inequalities involving products of these numbers.
Number Inequality
An inequality is a relation that holds between two different numbers when they are not equal. It tells us if one number is greater than, less than, or, in some cases, not less than (i.e., greater than or equal to) another number. Inequalities are fundamental aspects of mathematics and are as important as the concept of equality.

The symbols used in inequalities are \(<, >, \leq, \geq\) representing 'less than', 'greater than', 'less than or equal to', and 'greater than or equal to' respectively. Understanding how to manipulate inequalities is vital when solving problems, particularly when it comes to dividing both sides by a positive number, which maintains the integrity of the relationship, unlike dividing by a negative number, which flips the inequality.
Mathematical Proof
In mathematics, a proof is a logical argument that establishes the truth of a theorem or statement beyond any doubt, using a series of deductive steps. Proofs often involve assumption, deduction, and conclusion stages. When working with proofs related to inequalities in the context of natural numbers, one key point is to ensure that each manipulation of the inequality preserves the relationship between the quantities involved.

In the case of natural numbers, since we know the numbers involved are inherently positive, dividing across an inequality by a natural number does not affect the direction of the inequality. This fact is used to simplify the given problem and arrive at a conclusion that is logically sound and indisputably correct, provided all steps adhere to mathematical laws and definitions.

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

Find the solution set of the given inequality. $$ (x-2) / x<3 $$

Show that if \(a_{1}, a_{2}, \ldots, a_{5}\) is a sequence of 5 terms, then \(\prod_{i=1}^{5} a_{i}=0\) if and only if at least one term of the sequence is zero.

Show that \(a(b+c+d)=a b+a c+a d\), giving reasons for each step.

Is it possible to make addition and multiplication tables so that the four elements \(0,1,2,3\) form the elements of a field? Prove your statement. [Hint: In the multiplication table each row, other than the one consisting of zeros, must contain the symbols \(0,1,2,3\) in some order.]

Use the Principle of mathematical induction to establish the given formula. $$ \sum_{i=1}^{n}(1 / i(i+1))=n /(n+1) $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

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.