Firstly, take notice that we cannot add brackets of cross product, which are totally different.
And, take notice of the on. If on N, means aRb, it’s a pair (a,b). If on $\mathbb{N} \times \mathbb{N}$, its (a,b)R(c,d), namely ((a,b),(c,d)).
Welcome to point out the mistakes and faults!