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
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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c836437-a2a4-49ad-8637-6abf33735fd4%2Fde58e790-d193-47b7-acf9-538b16b7852b%2Ffodxi2e_processed.jpeg&w=3840&q=75)
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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