group theory – If the subsemigroup N of Z with respect to operation of addition, then show that R^N≠R^Z intersection (N×N).

I have tried to attempt it. But I don’t know if it is true.
Proof by contradiction
On the contrary, suppose the subsemigroup N of Z with respect to operation of addition. To show the relation on N is equal to the operation of intersection between the relation R on Z and N×N.

Note: N, i.e., the set of all natural numbers. Z, i.e., the set of all integers. R, i.e., is the relation. R^N, i.e., the relation on N and R^Z,i.e., the relation on Z and N×N, i.e., the cartesian product of N into N and it is containing ordered pairs.

I need its proof so that I can come to know its construction of proof and how the given information is used.