Solution

Since
$R\subset \mathbb{N}\times \mathbb{N}$,

the relation $R$ can have at most countably infinite elements.

Now, suppose for some $a\in \mathbb{N}$, the set $R_a$ is infinite.

Since $R_a\subseteq \mathbb{N}$, it is countably infinite.

Also, for every element $b\in R_a$, the ordered pair $(a,b)$ belongs to $R$.

So, $R$ is also infinite. Since $R\subset \mathbb{N}\times \mathbb{N}$, $R$ is countably infinite.

Therefore, both $R_a$ and $R$ have the same cardinality.