A set $A$ is given and a relation $R$ is defined on $A$. Determine whether each is an equivalence relation. Be sure to prove your answer.

Question

Given $A=(x\in \Bbb R :x>0)$, we define $xRy \iff y/2 \leq x \leq 2y.$

Okay, there isn't any $x$ that satisfy this, but which equivalence relation is this? I think it's symmetric, but how do I prove it?

Answer 1

$xRy \implies y/2 \leq x \leq 2y \implies y/2 \leq x$ and $ x \leq 2y $ which can then be algebraically manipulated into $y \leq 2x$ and $ x/2 \leq y $. Recombining $ x/2 \leq y \leq 2x \implies yRx$