Finding the relation for the partitions { 2x } and { 2x + 1 }

Question

I have to find an equivalence relation in the set of natural numbers which has the two partitions

{ 2x } and { 2x + 1 }

My first thought was

R = { (x,y) in N² : 2 | x + y }

I assume that this is incorrect, because the set is N² and N.

Then I thought about the following:

R = { x in N : ∃ y in N : 2 | x + y }

Is that a correct way to define the relation with the partitions described above?

Answer 1

Your first thought is correct. A binary relation in $\mathbb N$ is supposed to be a subset of $\mathbb N^2$, so having a set of pairs of numbers is exactly what you need to write here.

Answer 2

Your first relation works. Can you prove it?

(You went to your second relation because you were concerned about $\Bbb N^2$ being the wrong set, but remember that a relation on a set $A$ is a subset of $A\times A$, so you actually do want $R\subseteq\Bbb N^2$.)

Answer 3

R = {(x,y) : 2 | (x - y)}

(x,y) is in R if and only if x and y are either both even or both odd.