Problem 42 - Discrete Mathematics with Applications

Question

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

Step-by-Step Solution

Answer

We are given \(a+c

Step 1: Rewrite the inequality

Given the inequality \(a+c

Step 2: Find x

Subtract \(c\) from both sides of the inequality \(a+c

Step 3: Show that x is positive

Since \(a < b\), then the difference between them, \(x = b - a\), must be a positive real number because \(b\) is greater than \(a\).

Step 4: Conclusion

We've expressed the inequality \(a

Key Concepts

Real Numbers

Real numbers are an invaluable part of mathematics encompassing all the numbers we are familiar with from everyday life: integers, fractions, and irrational numbers like \(\pi\) and \(\sqrt{2}\). They can be visualized as points on an endless number line.
Understanding real numbers is essential because they form the foundation for most mathematical concepts. In the context of inequalities, they allow us to discuss how values compare and interact.

• Real numbers include positive numbers, negative numbers, and zero.
• They can be both rational (like \(\frac{1}{2}, 4,\) and \(0.75\)) and irrational (like \(\sqrt{3}\), \(\pi\)).

Real numbers are crucial when expressing inequalities, as these expressions often require subtracting or adding values on the number line to establish their relationships.

Mathematical Proof

A mathematical proof is a logical argument that verifies the truth of a given statement or theorem. In our exercises involving inequalities, proofs help show that our logical deductions hold true consistently.
For inequalities, proofs often involve rearranging terms or properties like adding the same value to each side without flipping the inequality's direction.

• Proofs can be done by direct arguments as we see in our problem; simplifying \(a + c < b + c\) to \(a < b\).
• Using inequalities involves demonstrating properties such as transitivity, if \(a < b\) and \(b < c\), then \(a < c\).

When you establish a relation like \(a + x = b\) with \(x\) being a positive number, you convincingly show \(a < b\), adhering firmly to the inequality principle.

Positive Numbers

Positive numbers lie to the right of zero on the number line and are greater than zero. In mathematics, they play a critical role in verifying the truth of propositions involving inequalities.
Whenever you express an inequality such as \(a < b\) with \(a + x = b\) for a positive \(x\), it means that \(x\) is what shifts \(a\) up to \(b\).

• Positive numbers are always used to increase values because their addition results in a value larger than the original.
• The concept of positive numbers assures that when \(a + x = b\), we can deduce \(a < b\).

By recognizing that \(x = b - a\) is positive, you confirm \(a < b\), cementing a foundational concept in solving inequalities.

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Problem 41

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Problem 42

Other exercises in this chapter

Problem 41

Let UD = set of integers, \(P(x, y): x\) is a multiple of \(y,\) and \(Q(x, y): x \geq y\) Determine the truth value of each proposition. $$(\exists x) \mathrm{

View solution

Problem 41

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

View solution

Problem 42

Determine whether or not the assignment statement \(x \leftarrow x+1\) will be executed in each sequence of statements, where \(i \leftarrow 2, j \leftarrow 3,

View solution

Problem 42

Use De Morgan's laws to verify each. (Hint: \(p \rightarrow q \equiv \sim p \vee q\) ). $$\sim(\sim p \wedge \sim q) \equiv p \vee q$$

View solution

Key Concepts

Real Numbers

Real numbers are an invaluable part of mathematics encompassing all the numbers we are familiar with from everyday life: integers, fractions, and irrational numbers like \(\pi\) and \(\sqrt{2}\). They can be visualized as points on an endless number line.
Understanding real numbers is essential because they form the foundation for most mathematical concepts. In the context of inequalities, they allow us to discuss how values compare and interact.

• Real numbers include positive numbers, negative numbers, and zero.
• They can be both rational (like \(\frac{1}{2}, 4,\) and \(0.75\)) and irrational (like \(\sqrt{3}\), \(\pi\)).

Real numbers are crucial when expressing inequalities, as these expressions often require subtracting or adding values on the number line to establish their relationships.

Mathematical Proof

A mathematical proof is a logical argument that verifies the truth of a given statement or theorem. In our exercises involving inequalities, proofs help show that our logical deductions hold true consistently.
For inequalities, proofs often involve rearranging terms or properties like adding the same value to each side without flipping the inequality's direction.

• Proofs can be done by direct arguments as we see in our problem; simplifying \(a + c < b + c\) to \(a < b\).
• Using inequalities involves demonstrating properties such as transitivity, if \(a < b\) and \(b < c\), then \(a < c\).

When you establish a relation like \(a + x = b\) with \(x\) being a positive number, you convincingly show \(a < b\), adhering firmly to the inequality principle.

Positive Numbers

Positive numbers lie to the right of zero on the number line and are greater than zero. In mathematics, they play a critical role in verifying the truth of propositions involving inequalities.
Whenever you express an inequality such as \(a < b\) with \(a + x = b\) for a positive \(x\), it means that \(x\) is what shifts \(a\) up to \(b\).

• Positive numbers are always used to increase values because their addition results in a value larger than the original.
• The concept of positive numbers assures that when \(a + x = b\), we can deduce \(a < b\).

By recognizing that \(x = b - a\) is positive, you confirm \(a < b\), cementing a foundational concept in solving inequalities.