Prove that $R$ is an equivalence relation: $aRb$ iff $2a+3b$ is divisible by $5$. Here $a,b\in \mathbb{Z}$ (set of integers).
I can prove that $R$ is reflexive and transitive. How to prove it's symmetric?
divisibilityequivalence-relationsrelations
Prove that $R$ is an equivalence relation: $aRb$ iff $2a+3b$ is divisible by $5$. Here $a,b\in \mathbb{Z}$ (set of integers).
I can prove that $R$ is reflexive and transitive. How to prove it's symmetric?
Best Answer
$2a+3b$ is divisible by $5$ if and only if $2a+3b-5b=2a-2b$ is divisible by $5$, if and only if $a-b$ is divisible by $5$ (since $5$ is prime). Now it should be easy.
A direct proof.
For $a\in\mathbb{Z}$, $2a+3a=5a$ is divisible by $5$
Suppose $2a+3b=5h$; then $-2a-3b=-5h$, so $5(a+b)-2a-3b=5(a+b-h)$ and, finally $3b+2a=5(a+b-h)$
Suppose $2a+3b=5h$ and $2b+3c=5k$; then $2a+3b+2b+3c=5(h+k)$, so…