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Define a relation R on the real numbers as follows: x Ry if and only if [x-y] = 0. Is R an equivalence relation? Prove or disprove.

Define a relation R on the real numbers as follows: x Ry if and only if [x-y] = 0. Is R an equivalence relation? Prove or disprove.

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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6. The *nearest integer function* is the name of the function that we associate with “rounding” to the nearest integer. For a given real number \( x \), the nearest integer function is usually denoted by \([x]\). Do not confuse this with the notation for “equivalence class” which also uses brackets. Here are some examples: \[ [2.791] = 3, \quad [0.257] = 0, \quad [5/3] = 2, \quad [7.5] = 8, \quad [-5.49] = -5. \] (Notice, it is convention to round something like 7.5 *up* to 8, even though it is exactly in the middle of 7 and 8). Define a relation \( R \) on the real numbers as follows: \( x \, R \, y \) if and only if \([x - y] = 0\). Is \( R \) an equivalence relation? Prove or disprove.
Transcribed Image Text:6. The *nearest integer function* is the name of the function that we associate with “rounding” to the nearest integer. For a given real number \( x \), the nearest integer function is usually denoted by \([x]\). Do not confuse this with the notation for “equivalence class” which also uses brackets. Here are some examples: \[ [2.791] = 3, \quad [0.257] = 0, \quad [5/3] = 2, \quad [7.5] = 8, \quad [-5.49] = -5. \] (Notice, it is convention to round something like 7.5 *up* to 8, even though it is exactly in the middle of 7 and 8). Define a relation \( R \) on the real numbers as follows: \( x \, R \, y \) if and only if \([x - y] = 0\). Is \( R \) an equivalence relation? Prove or disprove.
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