Solution to Let R be a relation on the set of all non-negative integers defined by aRb … - Sikademy
Author Image

Archangel Macsika

Let R be a relation on the set of all non-negative integers defined by aRb if and only if a3 - b3 is divisible by 6. Then

The Answer to the Question
is below this banner.

Can't find a solution anywhere?

NEED A FAST ANSWER TO ANY QUESTION OR ASSIGNMENT?

Get the Answers Now!

You will get a detailed answer to your question or assignment in the shortest time possible.

Here's the Solution to this Question

Complete \space Question \space is \\ Let \space R \space be \space a \space relation \space on \space the \space set of \space all \space non-negative \space integers \space defined \space by \space aRb \space if \space and \space only \space if \space a ^{ 3 }- b ^{3 } \space is \space divisible \space by \space 6. Then \space relation \space is \space equivalence \space relation .\\ Solution \\ For \space check \space to \space equivalence \space relation, we \space will \space be \space check \\ (1)reflexive, (2) symmetric, and (3) transitive \\ (1) reflexive :-\\ If \space any \space element \space a \space is \space related \space to \space itself.then \space it \space called \space reflexive. Here \space a ^{3 } -a ^{ 3 } \space is \space equal \space to \space zero. And \space zero \space is \space divisible \space by \space 6 \\ So \space say \space that \space aRa. \\ (2)symmetric:- \\ If \space a \space related \space to \space b \space and \space b \space Also \space related \space to \space a, then \space it \space called \space symmetric .\\ Here \\ \space a ^{3 } -b^{ 3 } \space is \space divisible \space by \space 6 \iff \\ \space b ^{3 } -a^{ 3 } \space is \space divisible \space by \space 6 \iff \\ bRa \\ so \space it \space is \space reflexive. \\ (3) transitive:- \\ If \space aRb \space and \space bRc \implies aRc \\ It \space called \space transitive. \\ Here \\ Let \\ \space a ^{3 } -b^{ 3 } \space is \space divisible \space by \space 6 \space and \space\space b^{3 } -c^{ 3 } \space is \space divisible \space by \space 6 \\ \iff [( \space a ^{3 } -b^{ 3 } )+(b^3-c^3)]\space is \space divisible \space by \space 6 \\ \iff \space a ^{3 } -c^{ 3 } \space is \space divisible \space by \space 6 \\ aRc \\ So \space it \space is \space transitive. \\ Finally \space we \space say \space that \space relation \space is \space equivalence \space relation.


Related Answers

Was this answer helpful?

Join our Community to stay in the know

Get updates for similar and other helpful Answers

Question ID: mtid-5-stid-8-sqid-2918-qpid-1617