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11. 0<1 13. 0

11. 0<1 13. 0

Advanced Engineering Mathematics
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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Let F be an ordered field with at least two elements 0≠1. Then prove: 

 

**11.** \(0 < 1\) **13.** \(0 < a < b \implies 0 < b^{-1} < a^{-1}\) ### Explanation: 1. **Inequality Understanding:** - **Statement 11** establishes that zero is less than one, which is a fundamental concept in mathematics. 2. **Reverse Inequality with Reciprocals (Statement 13):** - Given that \(0 < a < b\), this implies that \(a\) and \(b\) are both positive numbers, and \(a\) is less than \(b\). - Consequently, when considering the reciprocals, it's stated that \(0 < b^{-1} < a^{-1}\). This reflects that the reciprocal of the larger number \(b\) is less than the reciprocal of the smaller number \(a\). ### Mathematical Insight: - This is a classic property of inequalities involving reciprocals. In essence, if you have two positive numbers where one is smaller than the other, the inequality reverses upon taking reciprocals. This can be particularly useful in problem-solving scenarios where dealing with fractions or rational expressions.
Transcribed Image Text:**11.** \(0 < 1\) **13.** \(0 < a < b \implies 0 < b^{-1} < a^{-1}\) ### Explanation: 1. **Inequality Understanding:** - **Statement 11** establishes that zero is less than one, which is a fundamental concept in mathematics. 2. **Reverse Inequality with Reciprocals (Statement 13):** - Given that \(0 < a < b\), this implies that \(a\) and \(b\) are both positive numbers, and \(a\) is less than \(b\). - Consequently, when considering the reciprocals, it's stated that \(0 < b^{-1} < a^{-1}\). This reflects that the reciprocal of the larger number \(b\) is less than the reciprocal of the smaller number \(a\). ### Mathematical Insight: - This is a classic property of inequalities involving reciprocals. In essence, if you have two positive numbers where one is smaller than the other, the inequality reverses upon taking reciprocals. This can be particularly useful in problem-solving scenarios where dealing with fractions or rational expressions.
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