Prove that the following statement is true for two natural nos. $m,n$ if and only $v(m) = v(n)$ where $v(k)$ is the highest power of $2$ dividing $k$. $\exists$ a set $A$ of positive integers such that $(i)$ $x,y \in \mathbb{N}, |x-y| = m \implies x \in A $ or $y \in A$ $(ii)$ $x,y \in \mathbb{N}, |x-y| = n \implies x \not\in A $ or $y \not\in A$