Problem 40 - Discrete Mathematics with Applications

Question

Let \(a, b,\) and \(c\) be any real numbers. Then \(a

Step-by-Step Solution

Answer

Using the given fact, we know that if \(a

Step 1: Understand the given fact

The given fact says that if \(a

Step 2: Apply the given fact to \(a<b\) and \(b<c\)

We are given that \(a

Step 3: Sum the two equations

Now, let's sum the two equations we found in step 2: - Equation 1: \(a+x=b\) - Equation 2: \(b+y=c\) Adding the two equations yields: \[a+x+b+y=c\]

Step 4: Rearrange the equation

Let's rearrange the equation from step 3: \[a+x+b+y=c\] We can rewrite this as: \[a+(x+y)=c\]

Step 5: Prove the transitive property

From step 4, we have \(a+(x+y)=c\). Since both \(x\) and \(y\) are positive real numbers, their sum \((x+y)\) is also a positive real number. Let's call this positive real number \(z\): \[a+z=c\] Now, applying the given fact in reverse, we can say that \(a

Key Concepts

Real Numbers

In mathematics, real numbers are an essential concept that encompasses a vast range of numbers we use in daily life. These include whole numbers, fractions, and irrational numbers.
One distinguishing feature of real numbers is that they extend along a continuous number line without any gaps.Real numbers can be categorized into two main types:

• Rational numbers: Numbers that can be expressed as the quotient of two integers, such as \( \frac{1}{2} \) or \( -3 \).
• Irrational numbers: Numbers that cannot be written as a simple fraction, like \( \pi \) or \( \sqrt{2} \).

Real numbers are indispensable in expressing measurements such as length, mass, or time. They form the backbone of many mathematical proofs and operations, including those involving inequalities, as seen in the transitive property of inequality.

Inequalities

Inequalities are mathematical expressions that show the relationship of one quantity being larger or smaller than another. They are represented using symbols such as:

• \( < \): Less than
• \( > \): Greater than
• \( \leq \): Less than or equal to
• \( \geq \): Greater than or equal to

An important property of inequalities is the transitive property, which states that if one quantity is less than a second, and the second is less than a third, then the first is less than the third.
For instance, if we have three real numbers \(a, b,\) and \(c\) such that \(a < b\) and \(b < c\), we can conclude that \(a < c\).
This principle is crucial for problem-solving, allowing for the logical step-by-step progression from known inequalities to conclusions without examining each pair of numbers directly.

Proof in Mathematics

Mathematical proofs serve as the means by which mathematicians establish the truth of statements beyond reasonable doubt. Proofs use logical reasoning to show that one truth leads inevitably to another. There are several methods of proof, but the most common one involves the direct proof, where assumptions lead through logical steps to the conclusion. Direct proofs follow a straightforward path from given facts or assumptions to the thing to be proved.
In proving the transitive property of inequalities, a direct proof involves using known facts about real numbers and inequalities.Here's how it works:

• Start with known conditions (e.g., \(a < b\) and \(b < c\)).
• Use the transitive property to deduce that \(a < c\).
• Each step must logically follow from the previous.

This kind of logical reasoning enables mathematicians to build large networks of theorems from fundamental axioms, ensuring reliability and validity in mathematical studies.

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Problem 40

Other exercises in this chapter

Problem 39

Let \(x, y,\) and \(z\) be any real numbers. Represent each sentence symbolically, where \(p: xx\).

View solution

Problem 40

Mark each sentence as true or false, where \(p, q,\) and \(r\) are arbitrary statements, \(t\) a tautology, and \(f\) a contradiction. $$\text { If } p \wedge q

View solution

Problem 40

Let \(x, y,\) and \(z\) be any real numbers. Represent each sentence symbolically, where \(p: x

View solution

Problem 41

Let \(a, b,\) and \(c\) be any real numbers. Then \(a

View solution

Key Concepts

Real Numbers

In mathematics, real numbers are an essential concept that encompasses a vast range of numbers we use in daily life. These include whole numbers, fractions, and irrational numbers.
One distinguishing feature of real numbers is that they extend along a continuous number line without any gaps.Real numbers can be categorized into two main types:

• Rational numbers: Numbers that can be expressed as the quotient of two integers, such as \( \frac{1}{2} \) or \( -3 \).
• Irrational numbers: Numbers that cannot be written as a simple fraction, like \( \pi \) or \( \sqrt{2} \).

Real numbers are indispensable in expressing measurements such as length, mass, or time. They form the backbone of many mathematical proofs and operations, including those involving inequalities, as seen in the transitive property of inequality.

Inequalities

Inequalities are mathematical expressions that show the relationship of one quantity being larger or smaller than another. They are represented using symbols such as:

• \( < \): Less than
• \( > \): Greater than
• \( \leq \): Less than or equal to
• \( \geq \): Greater than or equal to

An important property of inequalities is the transitive property, which states that if one quantity is less than a second, and the second is less than a third, then the first is less than the third.
For instance, if we have three real numbers \(a, b,\) and \(c\) such that \(a < b\) and \(b < c\), we can conclude that \(a < c\).
This principle is crucial for problem-solving, allowing for the logical step-by-step progression from known inequalities to conclusions without examining each pair of numbers directly.

Proof in Mathematics

Mathematical proofs serve as the means by which mathematicians establish the truth of statements beyond reasonable doubt. Proofs use logical reasoning to show that one truth leads inevitably to another. There are several methods of proof, but the most common one involves the direct proof, where assumptions lead through logical steps to the conclusion. Direct proofs follow a straightforward path from given facts or assumptions to the thing to be proved.
In proving the transitive property of inequalities, a direct proof involves using known facts about real numbers and inequalities.Here's how it works:

• Start with known conditions (e.g., \(a < b\) and \(b < c\)).
• Use the transitive property to deduce that \(a < c\).
• Each step must logically follow from the previous.

This kind of logical reasoning enables mathematicians to build large networks of theorems from fundamental axioms, ensuring reliability and validity in mathematical studies.