Let $R = \{(a, b) | b = 2^ka\}$, for some non-negative integer $k$ and this is a binary relation on the set of natural numbers $\Bbb{N}$. Show that $R$ is a partial order relation.
I know this relation is reflexive when $k=0$, but i cant prove that it is antisymmetric or transitive. please help... thank you
We have $ a=2^0a$, so $ aRa $. Now, if $ aRb $ and $ bRa $, then there are $ k, k'\in\mathbb {N}$ such that $ b=2^ka $ and $ a=2^{k'} b $. Then $ b=2^{k+k'} b$, which implies that $2^{k+k'}=1$, so $ k=k'=0$, and thus $ a=b $. Finally, if $ aRb $ and $ bRc $, then $ b=2^ka $ and $ c=2^mb $, for some $ k, m\in\mathbb {N} $, therefore $ c=2^{k+m} a $, and this means that $ aRc $. Hence, the relation is a partial order relation.