Show that the equality [r + y] = [2] + [y] is not valid for all real numbers r and y.
Show that the equality [r + y] = [2] + [y] is not valid for all real numbers r and y.
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:**Problem Statement:**
Show that the equality \(\lfloor x + y \rfloor = \lfloor x \rfloor + \lfloor y \rfloor\) is not valid for all real numbers \(x\) and \(y\).
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**Explanation:**
This problem asks you to demonstrate that the equation involving the floor function is not universally true. The floor function, \(\lfloor x \rfloor\), represents the greatest integer less than or equal to \(x\).
**Detailed Analysis:**
1. **Understanding the Floor Function:**
- \(\lfloor x + y \rfloor\): This is the largest integer less than or equal to the sum \(x + y\).
- \(\lfloor x \rfloor + \lfloor y \rfloor\): This represents the sum of the largest integers less than or equal to \(x\) and \(y\) separately.
2. **Counterexample:**
- Consider \(x = 1.5\) and \(y = 1.5\).
- \(\lfloor 1.5 \rfloor = 1\) and \(\lfloor 1.5 \rfloor = 1\).
- Thus, \(\lfloor x \rfloor + \lfloor y \rfloor = 1 + 1 = 2\).
- However, \(\lfloor x + y \rfloor = \lfloor 1.5 + 1.5 \rfloor = \lfloor 3 \rfloor = 3\).
This counterexample illustrates that the equality \(\lfloor x + y \rfloor = \lfloor x \rfloor + \lfloor y \rfloor\) does not hold in this case, proving that it is not valid for all real numbers \(x\) and \(y\).
Expert Solution
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Consider the provided question,
We first define ceiling function.
The ceiling function maps x to the least integer greater than or equal to x.
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