When defined on the set $N_1=\{1,2,3,\cdots\}$ of positive integers a
relation $\sim$ such that two positive integers $x$ and $y$ satisfy
$x\sim y$ if and only if $x/y=2^k$ for some integer $k$,
show that $\sim$ is an equivalence relation.
How do I approach proving that the relation holds? I understand that I need to prove that it is reflexive, symmetric, and transitive, but I don't entirely understand how to prove each case!
Best Answer
Reflexive
$\forall n \in N_1: \dfrac nn = 1 = 2^0$
$\therefore n \sim n$
Symmetric
$\forall x,y \in N_1: x \sim y \implies \dfrac xy = 2^k \implies \dfrac yx = 2^{-k} \implies y \sim x$
Transitive
$\forall x,y,z \in N_1: x \sim y \land y \sim z \implies \dfrac xy = 2^m \land \dfrac yz = 2^n \implies \dfrac xz = 2^{m+n} \implies x \sim z$