To show that $\left\{(a,b)\in \mathbb N^2\colon \exists k\in \mathbb N\left(b=2^ka\right)\right\}$ is a partial order

Question

Let R={(a,b)|b=2^k a} for some non-negative integer k} and this is a binary relation on the set of natural numbers N. Show that R is a partial order relation.

I am not sure of my answer..

For something to be partial order it should be Reflexive, Antisymmetric and Transitive. I am not sure if I have done the right way can someone verify please?

Answer 1

Your proof is correct.

The relation is quite a handy one as an example: it's a way of taking the set $\mathbb{N}$ and applying an order on it that looks like countably-many copies of $\mathbb{N}$. (Each copy is given by a set $a \times \{ 2^k : k \geq 0 \}$, with the "less than" order.)