5. For all integers n, if n is blue, then n2 is green. (a) What are the hypotheses of the statement? (b) What is the conclusion of the statement? (c) Prove the statement.
5. For all integers n, if n is blue, then n2 is green. (a) What are the hypotheses of the statement? (b) What is the conclusion of the statement? (c) Prove the statement.
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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Transcribed Image Text:**Mathematical Logic Exercise**
**Problem Statement:**
5. For all integers \( n \), if \( n \) is blue, then \( n^2 \) is green.
**Questions:**
(a) What are the hypotheses of the statement?
(b) What is the conclusion of the statement?
(c) Prove the statement.
**Explanation:**
This exercise explores conditional statements and logical reasoning in mathematics. The objective is to understand the hypothesis and conclusion in a given mathematical statement and to practice proving it using logical steps.
Transcribed Image Text:**Transcription for an Educational Website**
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### Problem Set
**6. Write each number as a ratio of two integers.**
(a) 6.2893
(b) \(2 \frac{4}{7} + \frac{4}{3}\)
(c) 0.243243243243243...
(d) 562.93045045045045045...
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**7. Assume that \(k\) is an integer. Explain why \(\frac{5k}{k^2 + 2}\) must be a rational number.**
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**Explanation of Concepts:**
In problem 6, the task is to express different kinds of numbers as fractions or ratios of two integers, which involves:
- Converting decimals to fractions.
- Summing mixed numbers with fractions.
- Identifying patterns in repeating decimals.
- Decomposing numbers with repeating properties in decimals.
In problem 7, the focus is on showing that a given expression is rational, which means it can be written as a fraction of two integers.
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*Note: No graphs or diagrams were present in this document.*
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