b. Determine whether the statement is true or false and justify your answer. Determine which of the statements in 15-19 are true and which are false. Prove each true statement directly from the definitions, and give a counterexample for each false statement. For a statement that is false, determine whether a small change would make it true. If so, make the change and prove the new statement. Follow the directions for writing proofs on page 173. 15. The product of any two rational numbers is a rational number.
b. Determine whether the statement is true or false and justify your answer. Determine which of the statements in 15-19 are true and which are false. Prove each true statement directly from the definitions, and give a counterexample for each false statement. For a statement that is false, determine whether a small change would make it true. If so, make the change and prove the new statement. Follow the directions for writing proofs on page 173. 15. The product of any two rational numbers is a rational number.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
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#15 please
![### Transcription and Explanation
#### Instructions
**Determine which of the statements in 15–19 are true and which are false. Prove each true statement directly from the definitions, and give a counterexample for each false statement. For a statement that is false, determine whether a small change would make it true. If so, make the change and prove the new statement. Follow the directions for writing proofs on page 173.**
#### Statement
15. The product of any two rational numbers is a rational number.
---
#### Problem
**29. Suppose \( a, b, \) and \( c \) are integers and \( x, y, \) and \( z \) are nonzero real numbers that satisfy the following equations:**
\[
\frac{xy}{x+y} = a \quad \text{and} \quad \frac{xz}{x+z} = b \quad \text{and} \quad \frac{yz}{y+z} = c
\]
**Is \( x \) rational? If so, express \( x \) as a ratio of two integers.**
#### Explanation
The page poses mathematical problems requiring verification and proof, focusing on rational numbers—a number that can be expressed as the quotient or fraction of two integers.
1. **Statement 15** questions the properties of rational numbers regarding multiplication.
2. **Problem 29** involves solving equations under given conditions to determine the rationality of \( x \) and, if applicable, expressing it as a ratio of two integers.
This exercise sharpens the ability to reason logically and understand arithmetic properties and operations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8ec5dee8-4b81-4c1e-b6ca-023c2626018b%2F45bb67c6-36f2-4ad1-84d0-9868eee70a69%2Fhcp5o3a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Transcription and Explanation
#### Instructions
**Determine which of the statements in 15–19 are true and which are false. Prove each true statement directly from the definitions, and give a counterexample for each false statement. For a statement that is false, determine whether a small change would make it true. If so, make the change and prove the new statement. Follow the directions for writing proofs on page 173.**
#### Statement
15. The product of any two rational numbers is a rational number.
---
#### Problem
**29. Suppose \( a, b, \) and \( c \) are integers and \( x, y, \) and \( z \) are nonzero real numbers that satisfy the following equations:**
\[
\frac{xy}{x+y} = a \quad \text{and} \quad \frac{xz}{x+z} = b \quad \text{and} \quad \frac{yz}{y+z} = c
\]
**Is \( x \) rational? If so, express \( x \) as a ratio of two integers.**
#### Explanation
The page poses mathematical problems requiring verification and proof, focusing on rational numbers—a number that can be expressed as the quotient or fraction of two integers.
1. **Statement 15** questions the properties of rational numbers regarding multiplication.
2. **Problem 29** involves solving equations under given conditions to determine the rationality of \( x \) and, if applicable, expressing it as a ratio of two integers.
This exercise sharpens the ability to reason logically and understand arithmetic properties and operations.
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