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Chapter 3: Problem 14

(a) Verify that the triangle inequality is true for several different real numbers \(x\) and \(y .\) Be sure to have some examples where the real numbers are negative. (b) Explain why the following proposition is true: For each real number \(r\), \(-|r| \leq r \leq|r|\) (c) Now let \(x\) and \(y\) be real numbers. Apply the result in Part (14b) to both \(x\) and \(y\). Then add the corresponding parts of the two inequalities to obtain another inequality. Use this to prove that \(|x+y| \leq|x|+|y|\)

Short Answer

Expert verified
We verified the triangle inequality for different real numbers $x$ and $y$ with examples. Additionally, we proved that for any real number $r$, $-|r| \leq r \leq |r|$. Finally, by applying this proposition to both $x$ and $y$, we added the corresponding parts of the two inequalities and used this new inequality to prove the triangle inequality: $|x+y| \leq |x| + |y|$.

Step by step solution

01

Choose real numbers for x and y

Choose a few pairs of real numbers for x and y, with a mix of positive and negative numbers. Example 1: x = 2, y = 3 Example 2: x = -2, y = 3 Example 3: x = -2, y = -3
02

Verify the inequality for chosen examples

For each chosen example, verify the triangle inequality \(|x + y| \leq |x| + |y|\): Example 1: \[|2+3|=5 \leq 2+3=|2|+|3|\] Example 2: \[|-2+3|=1 \leq 2+3=|-2|+|3|\] Example 3: \[|-2 + (-3)|=|-5|=5 \leq 2 + 3=|-2|+|-3|\] Thus, the triangle inequality holds for each of the chosen examples. (b) Explaining the given proposition
03

Understanding the proposition

The given proposition states that for each real number r, \(-|r| \leq r \leq |r|\). This is true because the absolute value of r, denoted |r|, represents the distance of r from 0. The absolute value of r is always non-negative, and r can be either positive, negative, or zero. The inequality states the lower and upper bounds of possible values for r.
04

Proving the proposition

For r 鈮� 0: \[r = |r| \geq 0\] Hence, \(-|r| \leq 0 \leq r\] For r < 0: \[|r|=-r\] Hence, \(-|r| = -(-r) = r\] So, \(-|r|\leq r \leq |r|\) In both cases, the proposition holds. Therefore, it is true for all real numbers. (c) Using Part (b) result to prove the triangle inequality
05

Apply the result from Part (b) to x and y

From Part (b), we know that \(-|x| \leq x \leq |x|\) and \(-|y| \leq y \leq |y|\).
06

Add the corresponding parts of the inequalities

Add the left sides and right sides of the two inequalities respectively: \[(-|x|) + (-|y|) \leq x + y \leq |x|+|y|\]
07

Use this inequality to prove the triangle inequality

From the inequality obtained in Step 6, we know that \(-(|x|+|y|) \leq x+y \leq |x|+|y|\). Now, because: \[|x+y|\geq 0\] \[|x+y| \geq -(|x|+|y|)\] Combining this with the Step 6 inequality, we have: \[0\leq |x+y| \geq -(|x| + |y|) \leq x + y \leq |x| + |y|\] Hence, we've proven that \(|x+y| \leq |x| + |y|\), which is the triangle inequality.

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Key Concepts

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

Absolute Value
The absolute value of a real number can be thought of as its distance from zero on the number line, without considering direction. Whenever you see an expression like \( |x| \) where \( x \) is any real number, it represents how far \( x \) is from zero. So, negative numbers become positive because distance is always non-negative.

For instance: \( |-3| = 3 \) and \( |4| = 4 \) because both 3 and 4 are three and four units away from zero, respectively. This concept is crucial when dealing with the triangle inequality because it entails adding the absolute values of numbers, which could be positive or negative, but considering them as distances which are always positive.
Real Numbers
Real numbers include all the numbers on the number line; they can be either rational or irrational, positive, negative, or zero. They represent a continuous value, which means that between any two real numbers, there exists another real number.

Familiarizing yourself with real numbers is essential for understanding concepts such as the triangle inequality because it applies to all real numbers without exception. Whether you're dealing with the square root of two, pi, or -5, they all fall under the umbrella of real numbers and abide by the laws and properties associated with them.
Inequalities
Inequalities express the relative size or order of two values. They are mathematical statements that compare expressions and tell us if one number is larger, smaller, or equal to another number, with the aid of signs like \( > \) (greater than), \( < \) (less than), \( \geq \) (greater than or equal to), and \( \leq \) (less than or equal to).

When dealing with the absolute values of real numbers, inequalities help us determine the boundaries within which these numbers exist. For example, the statement \( -|r| \leq r \leq |r| \) with \( r \) being a real number, gives us a concrete way to understand the range within which \( r \) can be found, taking into account its absolute value.
Mathematical Reasoning
Mathematical reasoning is a critical skill that involves making logical deductions and crafting mathematical proofs. Being able to logically deduce properties or laws such as the triangle inequality not only shows a profound understanding of mathematical concepts but also demonstrates the ability to apply such reasoning in varied situations.

In the context of our exercise, mathematical reasoning is employed in breaking down the statement of the triangle inequality and linking it to the properties of absolute values and real numbers. Through a sequence of logical steps, we establish a connection between known truths (the properties of absolute values) and the proposition we aim to prove. This process underscores the value of a comprehensive, step-by-step approach to solving complex mathematical problems.

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Most popular questions from this chapter

For each of the following, use a counterexample to prove the statement is false. * (a) For each odd natural number \(n\), if \(n>3\), then 3 divides \(\left(n^{2}-1\right)\). (b) For each natural number \(n,\left(3 \cdot 2^{n}+2 \cdot 3^{n}+1\right)\) is a prime number. (c) For all real numbers \(x\) and \(y, \sqrt{x^{2}+y^{2}}>2 x y\). (d) For each integer \(a\), if 4 divides \(\left(a^{2}-1\right)\), then 4 divides \((a-1)\).

In Preview Activity \(2,\) we proved that if \(n\) is an integer, then \(n^{2}+n\) is an even integer. We define two integers to be consecutive integers if one of the integers is one more than the other integer. This means that we can represent consecutive integers as \(m\) and \(m+1,\) where \(m\) is some integer.

Using a Logical Equivalency. Consider the following proposition: Proposition. For all integers \(a\) and \(b,\) if 3 does not divide \(a\) and 3 does not divide \(b,\) then 3 does not divide the product \(a \cdot b\). (a) Notice that the hypothesis of the proposition is stated as a conjunction of two negations (" 3 does not divide \(a\) and 3 does not divide \(b\) "). Also, the conclusion is stated as the negation of a sentence ("3 does not divide the product \(a \cdot b\) "). This often indicates that we should consider using a proof of the contrapositive. If we use the symbolic form \((\neg Q \wedge \neg R) \rightarrow \neg P\) as a model for this proposition, what is \(P,\) what is \(Q,\) and what is \(R ?\) (b) Write a symbolic form for the contrapositive of \((\neg Q \wedge \neg R) \rightarrow \neg P\). (c) Write the contrapositive of the proposition as a conditional statement in English. We do not yet have all the tools needed to prove the proposition or its contrapositive. However, later in the text, we will learn that the following proposition is true. Proposition \(\mathbf{X}\). Let \(a\) be an integer. If 3 does not divide \(a\), then there exist integers \(x\) and \(y\) such that \(3 x+a y=1\) (d) i. Find integers \(x\) and \(y\) guaranteed by Proposition \(\mathrm{X}\) when \(a=5\). ii. Find integers \(x\) and \(y\) guaranteed by Proposition \(\mathrm{X}\) when \(a=2\). iii. Find integers \(x\) and \(y\) guaranteed by Proposition \(\mathrm{X}\) when \(a=-2\). (e) Assume that Proposition \(\mathrm{X}\) is true and use it to help construct a proof of the contrapositive of the given proposition. In doing so, you will most likely have to use the logical equivalency \(P \rightarrow(Q \vee R) \equiv\) \((P \wedge \neg Q) \rightarrow R\)

The purpose of this exploration is to investigate the possibilities for which integers cannot be the sum of the cubes of two or three integers. (a) If \(x\) is an integer, what are the possible values (between 0 and 8 , inclusive) for \(x^{3}\) modulo \(9 ?\) (b) If \(x\) and \(y\) are integers, what are the possible values for \(x^{3}+y^{3}\) (between 0 and 8 , inclusive) modulo \(9 ?\) (c) If \(k\) is an integer and \(k \equiv 3(\bmod 9), \operatorname{can} k\) be equal to the sum of the cubes of two integers? Explain. (d) If \(k\) is an integer and \(k \equiv 4(\bmod 9), \operatorname{can} k\) be equal to the sum of the cubes of two integers? Explain. (e) State and prove a theorem of the following form: For each integer \(k\), if (conditions on \(k\) ), then \(k\) cannot be written as the sum of the cubes of two integers. Be as complete with the conditions on \(k\) as possible based on the explorations in Part (b). (f) If \(x, y,\) and \(z\) are integers, what are the possible values (between 0 and 8 , inclusive) for \(x^{3}+y^{3}+z^{3}\) modulo \(9 ?\) (g) If \(k\) is an integer and \(k \equiv 4(\bmod 9),\) can \(k\) be equal to the sum of the cubes of three integers? Explain. (h) State and prove a theorem of the following form: For each integer \(k\), if (conditions on \(k\) ), then \(k\) cannot be written as the sum of the cubes of three integers. Be as complete with the conditions on \(k\) as possible based on the explorations in Part (f).

(a) Use cases based on congruence modulo 3 and properties of congruence to prove that for each integer \(n, n^{3}=n(\mathrm{mod} 3)\). (b) Explain why the result in Part (a) proves that for each integer \(n, 3\) divides \(\left(n^{3}-n\right) .\) Compare this to the proof of the same result in Proposition 3.27
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